微积分

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约定

$\lim\limits_{x⇝x_0 \ y⇝y_0}$表示$x⇝x_0$与$y⇝y_0$各自独立发生,极限符号同侧的两个极限过程同时发生。

$\lim\limits_{n⇝∞^{+} }^{Δx⇝0}$表示因为$Δx⇝0$而导致$n⇝∞^{+}$,极限符号上下的两个极限过程同时发生。

$\lim\limits_{y⇝y_0} \lim\limits_{x⇝x_0}$表示$x⇝x_0$领先于$y⇝y_0$发生,两个极限符号的极限过程并非同时发生。

极值点

任意实数的绝对值,为其与实数轴上原点之间的距离。

$ x - x_0 ≤ δ$ $x - x_0 ∈ [-δ, 0⁻] ∪ [0⁺, +δ]$ $[-δ ≤ x - x_0 ≤ 0⁻] ∨ [ 0⁺ ≤ x - x_0 ≤ +δ]$ $[-δ ≤ x - x_0] ∧ [x - x_0 ≤ +δ]$
$ x - x_0 ≥ δ$ $x - x_0 ∈ [∞⁻, -δ] ∪ [+δ, ∞⁺]$ $[∞⁻ ≤ x - x_0 ≤ -δ] ∨ [+δ ≤ x - x_0 ≤ ∞⁺]$ $[x - x_0 ≤ -δ] ∨ [x - x_0 ≥ +δ]$

无穷大可视为扩充的实数轴上向其两端无穷远处延伸出的一点,扩充的实数轴在其两端的无穷大点处形成封闭的圆环,实数轴上除无穷大点外均按线性序排列。

无穷大的绝对值为正无穷大,负无穷大与正无穷大互为相反数。但在通常意义中,不考虑扩充的实数轴以及无穷大点,而视实数轴为以原点为中点的直线数轴。

无穷大点与原点连线将实数轴划分为负半实数轴与正半实数轴,负无穷大可视为从负半实数轴趋近于无穷大点,正无穷大可视为从正半实数轴趋近于无穷大点。

无穷大 $∞$   $\dfrac{1}{0} = ∞$ $\dfrac{1}{∞} = 0$ $\lim\limits_{n⇝∞⁺} (-1)^n · n ⇝ ∞$ $x_0 + ∞ \mathop{=}\limits_{x_0≠∞} ∞$ $x_0 - ∞ \mathop{=}\limits_{x_0≠∞} ∞$ $(+ x_0 ) · ∞ \mathop{=}\limits_{x_0≠0} ∞$ $(- x_0 ) · ∞ \mathop{=}\limits_{x_0≠0} ∞$    
正无穷大 $∞⁺$ $ = ∞⁺$ $\dfrac{1}{0⁺} = ∞⁺$ $\dfrac{1}{∞⁺} = 0⁺$ $\lim\limits_{n⇝∞⁺} (+1) · n ⇝ ∞⁺$ $x_0 + ∞⁺ \mathop{=}\limits_{x_0≠∞⁻} ∞⁺$ $x_0 - ∞⁺ \mathop{=}\limits_{x_0≠∞⁺} ∞⁻$ $(+ x_0 ) · ∞⁺ \mathop{=}\limits_{x_0≠0} ∞⁺$ $(- x_0 ) · ∞⁺ \mathop{=}\limits_{x_0≠0} ∞⁻$
负无穷大 $∞⁻$ $∞⁻ = -∞⁺$ $\dfrac{1}{0⁻} = ∞⁻$ $\dfrac{1}{∞⁻} = 0⁻$ $\lim\limits_{n⇝∞⁺} (-1) · n ⇝ ∞⁻$ $x_0 + ∞⁻ \mathop{=}\limits_{x_0≠∞⁺} ∞⁻$ $x_0 - ∞⁻ \mathop{=}\limits_{x_0≠∞⁻} ∞⁺$ $(+ x_0 ) · ∞⁻ \mathop{=}\limits_{x_0≠0} ∞⁻$ $(- x_0 ) · ∞⁻ \mathop{=}\limits_{x_0≠0} ∞⁺$    

点$x_0$的闭邻域$\mathrm{B}{X}(x_0, δ)$由属于区域$X$中的点构成,点$x_0$未必属于区域$X$。点$x_0$的任意闭邻域$\mathrm{B}{X}(x_0,δ)$与其对应的补邻域$\mathrm{B}_{X}^{¬}(x_0,δ)$之间没有交集。

孤立点是离散性质,聚敛点是极限性质。边界点要么为孤立点,要么为聚敛点。区域$X$的闭包集$\fbox{X}$由区域$X$本身以及其所有边界点共同构成。

最大值与最小值是全局性质与离散性质,极大值与极小值是局部性质与离散性质,上确界与下确界是极限性质与连续性质以及线性序性质。

上确界是所有上界的最小值,下确界是所有下界的最大值。上确界与下确界始终存在且唯一确定,因此上极限与下极限始终存在且唯一确定。

左闭邻域 $\mathrm{B}_{X}(x_0^{-},δ)$ $∀x∈X; [x∈\mathrm{B}_{X}(x_0^{-}, δ)] ⇔ [-δ ≤ x-x_0 ≤ 0]$      
右闭领域 $\mathrm{B}_{X}(x_0^{+},δ)$ $∀x∈X; [x∈\mathrm{B}_{X}(x_0^{+}, δ)] ⇔ [0 ≤ x-x_0 ≤ +δ]$      
闭邻域 $\mathrm{B}_{X}(x_0, δ)$ $∀x∈X; [ x∈\mathrm{B}_{X}(x_0, δ) ] ⇔ [ x - x_0 ≤ δ ]$ $\mathrm{B}{X}(x_0,δ) ⋂ \mathrm{B}{X}^{¬}(x_0,δ) = 𝟘$
补邻域 $\mathrm{B}_{X}^{¬}(x_0,δ)$ $∀x∉X; [x∈\mathrm{B}_{X}^{¬}(x_0, δ)] ⇔ [ x-x_0 ≤δ]$ $\mathrm{B}{X}(x_0,δ) ⋃ \mathrm{B}{X}^{¬}(x_0,δ) = \mathrm{B}_{𝟙}(x_0,δ)$
去心闭邻域 $\mathrm{\mathop{B} }_{X}^{∘}(x_0, δ)$ $∀x∈X; [x∈\mathrm{\mathop{B} }_{X}^{∘}(x_0, δ) ] ⇔ [0 < x - x_0 ≤ δ]$  
闭包集 $\fbox{X}$ $∀x_0; \left[ x_0∈\fbox{X} \right] ⇔ [ ∀δ>0; \mathrm{B}_{X}(x_0, δ) ≠ 𝟘 ]$ $\fbox{X}⊇X$    
聚敛点 $x_0∈\fbox{X}$ $∀δ>0; \mathrm{B}_{X}(x_0, δ) ≠ 𝟘$      
孤立点 $x_0∈X$ $∃δ>0; \mathrm{B}_{X}(x_0, δ) = \lbrace x_0 \rbrace$      
边界点 $x_0∈\fbox{X}$ $∀δ>0; [ \mathrm{B}{X}(x_0,δ) ≠ 𝟘 ] ∧ [ \mathrm{B}{X}^{¬}(x_0,δ) ≠ 𝟘 ]$      
内敛点 $x_0∈X$ $∃δ>0; \mathrm{B}_{X}(x_0, δ) ⊆ X$      
全闭区域 $[x_α, x_β]$ $[∀δ>0; \mathrm{B}(x_α, δ) ≠ 𝟘] ∧ [∀δ>0; \mathrm{B}(x_β, δ) ≠ 𝟘]$ $X = \fbox{X}$    
全开区域 $(x_α, x_β)$ $[∀δ>0; \mathrm{B}(x_α, δ) ≠ 𝟘] ∧ [∀δ>0; \mathrm{B}(x_β, δ) ≠ 𝟘]$ $X ≠ \fbox{X}$    
半闭区域 $[x_α, x_β)$ $x_α∈[x_α,x_β) ∧ x_β∉[x_α,x_β)$ $X ≠ \fbox{X}$    
半闭区域 $(x_α, x_β]$ $x_α∉(x_α,x_β] ∧ x_β∈(x_α,x_β]$ $X ≠ \fbox{X}$    
           
最大值 $m ≡ \max\limits_{m∈S} S$ $[ ∃m∈S;∀a∈S; m ≥ a ] ⇔ [ ∃m∈S;∀a∈S; m ≤ a ⇒ m = a ]$ 集合$S$为离散集    
最小值 $m ≡ \min\limits_{m∈S} S$ $[ ∃m∈S;∀a∈S; m ≤ a ] ⇔ [ ∃m∈S;∀a∈S; m ≥ a ⇒ m = a ]$ 集合$S$为离散集    
极大值 $m ≡ \max\limits_{m∈S⊆T} S$ $[ ∃m∈S;∀a∈S; m ≥ a ] ⇔ [ ∃m∈S;∀a∈S; m ≤ a ⇒ m = a ]$ 集合$S$为集合$T$的子集    
极小值 $m ≡ \min\limits_{m∈S⊆T} S$ $[ ∃m∈S;∀a∈S; m ≤ a ] ⇔ [ ∃m∈S;∀a∈S; m ≥ a ⇒ m = a ]$ 集合$S$为集合$T$的子集    
上确界 $s ≡ \sup S$ $[ ∃s∈\fbox{S};∀a∈S; a ≤ s ] ⇔ [∃s;∀o;∀a∈S; a ≤ s ∧ [a ≤ o ⇒ s ≤ o]]$ 集合$S$可为紧致集    
上确界 $s ≡ \sup S$ $[ ∃s;∀a∈S;∃e∈S; a ≤ s ∧ [ a < s ⇒ a < e ] ] ⇔ [ ∃s;∀a∈S;∀ε>0;∃e∈S; a ≤ s ∧ s - ε < e]$ 集合$S$可为紧致集    
下确界 $s ≡ \inf S$ $[ ∃s∈\fbox{S};∀a∈S; a ≥ s ] ⇔ [∃s;∀o;∀a∈S; a ≥ s ∧ [a ≥ o ⇒ s ≥ o]]$ 集合$S$可为紧致集    
下确界 $s ≡ \inf S$ $[ ∃s;∀a∈S;∃e∈S; a ≥ s ∧ [ a > s ⇒ a > e ] ] ⇔ [ ∃s;∀a∈S;∀ε>0;∃e∈S; a ≥ s ∧ s + ε > e]$ 集合$S$可为紧致集    
严格最大值 $m ≡ \max\limits_{m∈S} S$ $[ ∃m∈S;∀a∈S \backslash m; m > a ]$      
严格最小值 $m ≡ \min\limits_{m∈S} S$ $[ ∃m∈S;∀a∈S \backslash m; m < a ]$      
严格极大值 $m ≡ \max\limits_{m∈S⊆T} S$ $[ ∃m∈S;∀a∈S \backslash m; m > a ]$      
严格极小值 $m ≡ \min\limits_{m∈S⊆T} S$ $[ ∃m∈S;∀a∈S \backslash m; m < a ]$      
最大值 $f (x_0) ≡ \max\limits_{x∈X} f (x)$ $[ ∃x_0∈X;∀x∈X; f (x_0) ≥ f (x) ] ⇔ [ ∃x_0∈X;∀x∈X; f (x_0) ≤ f (x) ⇒ f (x_0) = f (x) ]$      
最小值 $f (x_0) ≡ \min\limits_{x∈X} f (x)$ $[ ∃x_0∈X;∀x∈X; f (x_0) ≤ f (x) ] ⇔ [ ∃x_0∈X;∀x∈X; f (x_0) ≥ f (x) ⇒ f (x_0) = f (x) ]$      
极大值 $f (x_0) ≡ \max\limits_{x∈X⊆Y} f (x)$ $[ ∃x_0∈X;∀x∈X⊆Y; f (x_0) ≥ f (x) ] ⇔ [ ∃x_0∈X;∀x∈X⊆Y; f (x_0) ≤ f (x) ⇒ f (x_0) = f (x) ]$      
极小值 $f (x_0) ≡ \min\limits_{x∈X⊆Y} f (x)$ $[ ∃x_0∈X;∀x∈X⊆Y; f (x_0) ≤ f (x) ] ⇔ [ ∃x_0∈X;∀x∈X⊆Y; f (x_0) ≥ f (x) ⇒ f (x_0) = f (x) ]$      
上确界 $s ≡ \sup\limits_{x∈X} f (x)$ $[ ∃s∈\fbox{f(x)}; ∀x∈X; f (x) ≤ s ] ⇔ [ ∃s;∀a;∀x∈X; f (x) ≤ s ∧ [ f (x) ≤ a ⇒ s ≤ a ] ]$      
上确界 $s ≡ \sup\limits_{x∈X} f (x)$ $[ ∃s;∀x∈X;∃y∈X; f (x) ≤ s ∧ [ f (x) < s ⇒ f (x) < f (y) ] ] ⇔ [ ∃s;∀x∈X;∀ε>0;∃y∈X; f (x) ≤ s ∧ s - ε < f (y) ]$      
下确界 $s ≡ \inf\limits_{x∈X} f (x)$ $[ ∃s∈\fbox{f(x)};∀x∈X; f (x) ≥ s ] ⇔ [ ∃s;∀a;∀x∈X; f (x) ≥ s ∧ [ f (x) ≥ a ⇒ s ≥ a ] ]$      
下确界 $s ≡ \inf\limits_{x∈X} f (x)$ $[ ∃s;∀x∈X;∃y∈X; f (x) ≥ s ∧ [ f (x) > s ⇒ f (x) > f (y) ] ] ⇔ [ ∃s;∀x∈X;∀ε>0;∃y∈X; f (x) ≥ s ∧ s + ε > f (y) ]$      
严格最大值 $f (x_0) ≡ \max\limits_{x∈X} f (x)$ $[ ∃x_0∈X;∀x∈X \backslash x_0; f (x_0) > f (x) ]$      
严格最小值 $f (x_0) ≡ \min\limits_{x∈X} f (x)$ $[ ∃x_0∈X;∀x∈X \backslash x_0; f (x_0) < f (x) ]$      
严格极大值 $f (x_0) ≡ \max\limits_{x∈X⊆Y} f (x)$ $[ ∃x_0∈X;∀x∈X \backslash x_0; f (x_0) > f (x) ]$      
严格极小值 $f (x_0) ≡ \min\limits_{x∈X⊆Y} f (x)$ $[ ∃x_0∈X;∀x∈X \backslash x_0; f (x_0) < f (x) ]$      
           
左侧函数值 $f(x_0-Δ)$        
右侧函数值 $f(x_0+Δ)$        
函数左极限 $f(x) {x_0^{-} } ≡ \lim\limits{x⇝x_0^-} f(x) ⇝ f_{x_0^-}$ $Ⅎx_0∈\fbox{X};∀ε>0;∃δ>0;∀x∈X; -δ ≤ x - x_0 < 0 ⇒ f(x) - f_{x_0^-} ≤ ε$
函数右极限 $f(x) {x_0^{+} } ≡ \lim\limits{x⇝x_0^+} f(x) ⇝ f_{x_0^+}$ $Ⅎx_0∈\fbox{X};∀ε>0;∃δ>0;∀x∈X; 0 < x - x_0 ≤ +δ ⇒ f(x) - f_{x_0^+} ≤ ε$
函数极限值 $f(x) {x_0} ≡ \lim\limits{x⇝x_0} f(x) ⇝ f_{x_0}$ $Ⅎx_0∈\fbox{X};∀ε>0;∃δ>0;∀x∈\mathrm{\mathop{B} }^{∘}(x_0,δ); f(x) - f_{x_0} ≤ ε$
函数极限值 $f(x) {x_0} ≡ \lim\limits{x⇝x_0} f(x) ⇝ f_{x_0}$ $\lim\limits_{x⇝x_0^-} f(x) ⇝ f_{x_0^-} = f_{x_0} = f_{x_0^+} ⇜ \lim\limits_{x⇝x_0^+} f(x)$    
极限值 $\lim\limits_{x⇝x_0} f(x) ⇝ ∞^{±},∞$ $\lim\limits_{x⇝x_0} \dfrac{1}{f(x)} ⇝ 0^{±},0$      
极限值 $\lim\limits_{x⇝∞^{±},∞} f(x) ⇝ f_{∞^{±} }$ $\lim\limits_{\frac{1}{x}⇝0^{±},0} f(x) ⇝ f_{∞^{±} }$      
上极限 $\varlimsup\limits_{x⇝x_0} f (x)$ $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{\mathop{B} }^{∘}(x_0,δ)} f(x)$      
下极限 $\varliminf\limits_{x⇝x_0} f (x)$ $\lim\limits_{δ⇝0} \inf\limits_{x∈\mathrm{B}^{∘}(x_0,δ)} f(x)$      
唯一极限值 $\varliminf\limits_{x⇝x_0} f(x) = \varlimsup\limits_{x⇝x_0} f(x)$ $\lim\limits_{δ⇝0} \inf\limits_{x∈\mathrm{B}^{∘}(x_0,δ)} f(x) = \lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{B}^{∘}(x_0,δ)} f(x)$      
           
单调递增 $f^{↗} (x)$ $[∀x_1,x_2∈X; [ x_1 < x_2 ] ⇒ [ f^{↗} (x_1) ≤ f^{↗} (x_2)]]$      
单调递减 $f^{↘} (x)$ $[∀x_1,x_2∈X; [ x_1 < x_2 ] ⇒ [ f^{↘} (x_1) ≥ f^{↘} (x_2)]]$      
严格单调递增 $f^{↑} (x)$ $[∀x_1,x_2∈X; [ x_1 < x_2 ] ⇒ [ f^{↑} (x_1) < f^{↑} (x_2)]]$      
严格单调递减 $f^{↓} (x)$ $[∀x_1,x_2∈X; [ x_1 < x_2 ] ⇒ [ f^{↓} (x_1) > f^{↓} (x_2)]]$      
$∀x∈[x_α, x_β]; f (x) ≤ ε$ $⇔$ $\sup\limits_{x∈[x_α,x_β]} f (x) ≤ ε$   $∃x∈[x_α, x_β]; f (x) > ε$ $⇔$ $\sup\limits_{x∈[x_α,x_β]} f (x) > ε$                
$∀x∈[x_α,x_β]; f (x) < ε$ $⇔$ $\sup\limits_{x∈[x_α,x_β]} f (x) < ε$   $∃x∈[x_α,x_β]; f (x) ≥ ε$ $⇔$ $\sup\limits_{x∈[x_α,x_β]} f (x) ≥ ε$                
$∀x∈(x_α,x_β); f (x) ≤ ε$ $⇔$ $\sup\limits_{x∈(x_α,x_β)} f (x) ≤ ε$   $∃x∈(x_α,x_β); f (x) > ε$ $⇔$ $\sup\limits_{x∈(x_α,x_β)} f (x) > ε$                
$∀x∈(x_α,x_β); f (x) < ε$ $⇒$ $\sup\limits_{x∈(x_α,x_β)} f (x) ≤ ε$ $\rlap{≡≡≡≡≡≡≡≡≡}\sup\limits_{x⇝x_α} f (x) = ε$ $∃x∈(x_α,x_β); f(x) ≥ ε$ $⇒$ $\sup\limits_{x∈(x_α,x_β)} f (x) ≥ ε$            
                                             
$∀x∈[x_α,x_β]; f(x) ≥ ε$ $⇔$ $\inf\limits_{x∈[x_α,x_β]} f(x) ≥ ε$   $∃x∈[x_α,x_β]; f(x) < ε$ $⇔$ $\inf\limits_{x∈[x_α,x_β]} f(x) < ε$                
$∀x∈[x_α,x_β]; f(x) > ε$ $⇔$ $\inf\limits_{x∈[x_α,x_β]} f(x) > ε$   $∃x∈[x_α,x_β]; f(x) ≤ ε$ $⇔$ $\inf\limits_{x∈[x_α,x_β]} f(x) ≤ ε$                
$∀x∈(x_α,x_β); f(x) ≥ ε$ $⇔$ $\inf\limits_{x∈(x_α,x_β)} f(x) ≥ ε$   $∃x∈(x_α,x_β); f(x) < ε$ $⇔$ $\inf\limits_{x∈(x_α,x_β)} f(x) < ε$                
$∀x∈(x_α,x_β); f(x) > ε$ $⇒$ $\inf\limits_{x∈(x_α,x_β)} f(x) ≥ ε$ $\rlap{≡≡≡≡≡≡≡≡}{\inf\limits_{x⇝x_α} f(x) = ε}$ $∃x∈(x_α,x_β); f(x) ≤ ε$ $⇒$ $\inf\limits_{x∈(x_α,x_β)} f(x) ≤ ε$            
                                             
$\sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) ≤ ε$ $⇔$ $\sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t > δ$   $\sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε$ $⇔$ $\sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t ≤ δ$

上确界与下确界的运算性质。

$\inf\limits_{x∈Y⊇X} f(x) ≤ \inf\limits_{x∈X} f(x) ≤ \sup\limits_{x∈X} f(x) ≤ \sup\limits_{x∈Y⊇X} f(x)$

$\sup\limits_{x∈X} f(x) = -\inf\limits_{x∈X} [-f(x)]$

$\inf\limits_{x∈X} f(x) = -\sup\limits_{x∈X} [-f(x)]$

$\sup\limits_{x∈X} [f(x) + g(x)] ≤ \sup\limits_{x∈X} f(x) + \sup\limits_{x∈X} g(x)$

$\inf\limits_{x∈X} [f(x) + g(x)] ≥ \inf\limits_{x∈X} f(x) + \inf\limits_{x∈X} g(x)$

$\varlimsup\limits_{x⇝x_0} [f(x) + g(x)] ≤ \varlimsup\limits_{x⇝x_0} f(x) + \varlimsup\limits_{x⇝x_0} g(x)$

$\varliminf\limits_{x⇝x_0} [f(x) + g(x)] ≥ \varliminf\limits_{x⇝x_0} f(x) + \varliminf\limits_{x⇝x_0} g(x)$

$∀x∈X; f(x) ≤ \sup\limits_{x∈X} f(x)$ $∀x∈X; -f(x) ≥ -\sup\limits_{x∈X} f(x)$ $\inf\limits_{x∈X} [-f(x)] = -\sup\limits_{x∈X} f(x)$ $\sup\limits_{x∈X} f(x) = -\inf\limits_{x∈X} [-f(x)]$
$∀x∈X; f(x) ≥ \inf\limits_{x∈X} f(x)$ $∀x∈X; -f(x) ≤ -\inf\limits_{x∈X} f(x)$ $\sup\limits_{x∈X} [-f(x)] = -\inf\limits_{x∈X} f(x)$ $\inf\limits_{x∈X} f(x) = -\sup\limits_{x∈X} [-f(x)]$
$∀x∈X; f(x) ≤ \sup\limits_{x∈X} f(x)$ $∀x∈X; [f(x) + g(x)] ≤ \sup\limits_{x∈X} f(x) + \sup\limits_{x∈X} g(x)$ $\sup\limits_{x∈X} [f(x) + g(x)] ≤ \sup\limits_{x∈X} f(x) + \sup\limits_{x∈X} g(x)$ $\varlimsup\limits_{x⇝x_0} [f(x) + g(x)] ≤ \varlimsup\limits_{x⇝x_0} f(x) + \varlimsup\limits_{x⇝x_0} g(x)$
$∀x∈X; f(x) ≥ \inf\limits_{x∈X} f(x)$ $∀x∈X; [f(x) + g(x)] ≥ \inf\limits_{x∈X} f(x) + \inf\limits_{x∈X} g(x)$ $\inf\limits_{x∈X} [f(x) + g(x)] ≥ \inf\limits_{x∈X} f(x) + \inf\limits_{x∈X} g(x)$ $\varliminf\limits_{x⇝x_0} [f(x) + g(x)] ≥ \varliminf\limits_{x⇝x_0} f(x) + \varliminf\limits_{x⇝x_0} g(x)$

典例:函数$f (x) = \sin x$在区间$\left( 0, \dfrac{π}{2} \right)$上的取值范围为$0 < f (x) < 1$,其下确界为$0$,其上确界为$1$。

典例:函数$f (x) = \dfrac{1}{x}$在区间$(0, ∞⁺)$上的取值范围为$0 < f (x)$,其下确界为$0$,其上确界为$∞⁺$。

典例:函数$f(x) = \sin x$在区间$(0, π)$上的下确界为$-1$且上确界为$+1$,函数$g(x) = -\sin x$在区间$(0, π)$上的下确界为$-1$且上确界为$+1$,函数$f(x) + g(x)$在区间$(0, π)$上为$0$。

函数极限的运算性质,若$\lim\limits_{x⇝x_0} f(x) ⇝ f_{x_0}$,且$\lim\limits_{x⇝x_0} g(x) ⇝ g_{x_0}$。

$\lim\limits_{x⇝x_0} [f(x) + g(x)] = \lim\limits_{x⇝x_0} f(x) + \lim\limits_{x⇝x_0} g(x) ⇝ f_{x_0} + g_{x_0}$

$\lim\limits_{x⇝x_0} [f(x) - g(x)] = \lim\limits_{x⇝x_0} f(x) - \lim\limits_{x⇝x_0} g(x) ⇝ f_{x_0} - g_{x_0}$

$\lim\limits_{x⇝x_0} [f(x) · g(x)] = \lim\limits_{x⇝x_0} f(x) · \lim\limits_{x⇝x_0} g(x) ⇝ f_{x_0} · g_{x_0}$

$\lim\limits_{x⇝x_0} \dfrac{f(x)}{g(x)} = \dfrac{\lim\limits_{x⇝x_0} f(x)}{\lim\limits_{x⇝x_0} g(x)} \mathop{⇝}\limits_{g_{x_0}≠0} \dfrac{f_{x_0} }{g_{x_0} }$

$\lim\limits_{x⇝x_0} \mathrm{Con} · f(x) = \mathrm{Con} · \lim\limits_{x⇝x_0} f(x) ⇝ \mathrm{Con} · f_{x_0}$

$⇓$ $[∀ε_1>0;∃δ_1>0; ∀x∈\mathrm{B}^{∘}(x_0,δ_1); f(x) - f_{x_0} ≤ ε_1] ∧ [∀ε_2>0;∃δ_2>0; ∀x∈\mathrm{B}^{∘}(x_0,δ_2); g(x) - g_{x_0} ≤ ε_2]$                                    
$⇓$ $∀ε>0;∃δ=\min\lbrace δ_1,δ_2 \rbrace;∀x∈\mathrm{B}^{∘}(x_0,δ); [f(x) + g(x)] - [f_{x_0} + g_{x_0}] f(x) - f_{x_0} + g(x) - g_{x_0} = ε_1 + ε_2 = ε$                                
$⇓$ $∀ε>0;∃δ=\min\lbrace δ_1,δ_2 \rbrace;∀x∈\mathrm{B}^{∘}(x_0,δ); [f(x) - g(x)] - [f_{x_0} - g_{x_0}] f(x) - f_{x_0} + g(x) - g_{x_0} = ε_1 + ε_2 = ε$                                
$⇓$ $∀ε>0;∃δ=\min\lbrace δ_1,δ_2 \rbrace;∀x∈\mathrm{B}^{∘}(x_0,δ); f(x) · g(x) - f_{x_0} · g_{x_0}] ≤ f(x) · g(x) - g_{x_0} + g_{x_0} · f(x) - f_{x_0} \sup f(x) · ε_2 + g_{x_0} · ε_1 = ε$                  
$⇓$ $∀ε>0;∃δ=\min\lbrace δ_1,δ_2 \rbrace;∀x∈\mathrm{B}^{∘}(x_0,δ); \left \dfrac{f(x)}{g(x)} - \dfrac{f_{x_0} }{g_{x_0} } \right = \left \dfrac{f(x) · g_{x_0} - f_{x_0} · g(x)}{g(x) · g_{x_0} } \right ≤ \dfrac{\left f(x) \right · g(x) - g_{x_0} + g_{x_0} · f(x) - f_{x_0} }{\left g(x) · g_{x_0} \right } ≤ \dfrac{ \sup f(x) · ε_2 + g_{x_0} · ε_1}{ \inf g(x) · g_{x_0} } = ε$

若函数$f(x)$在点$x_0$处有极限$\lim\limits_{x⇝x_0} f(x) ⇝ f_{x_0} = y_0$,且函数$g(y)$在点$y_0$处有极限$\lim\limits_{y⇝y_0} g(y) ⇝ g_{y_0}$,前提条件$f(x)≠f_{x_0}$不可忽略。

$\lim\limits_{x⇝x_0} g(f(x)) \mathop{====}\limits_{f_{x_0}=y_0}^{f(x)≠f_{x_0} } \lim\limits_{y⇝y_0} g(y) ⇝ g_{y_0}$

$⇓$ $∀ζ>0;∃ε>0;∀y; y∈\mathrm{B}^{∘}(y_0,ε) ⇒ g(y)∈\mathrm{B}^{∘}(g_{y_0},ζ)$  
$⇓$ $∀ε>0;∃δ>0;∀x; x∈\mathrm{B}^{∘}(x_0,δ) ⇒ f(x)∈\mathrm{B}^{∘}(y_0,ε)$ $f_{x_0} = y_0$
$⇓$ $∀ζ>0;∃ε>0;∃δ>0;∀x; x∈\mathrm{B}^{∘}(x_0,δ) ⇒ f(x)∈\mathrm{B}^{∘}(y_0,ε) ⇒ g(f(x))∈\mathrm{B}^{∘}(g_{y_0},ζ)$ $f(x) ≠ f_{x_0}$
$⇓$ $∀ζ>0;∃ε>0;∀y=f(x); y∈\mathrm{B}^{∘}(y_0,ε) ⇒ g(y)∈\mathrm{B}^{∘}(g_{y_0},ζ)$  

反例:函数$g(y) = \mathop{0}\limits_{y=0};\mathop{1}\limits_{y≠0}$,且函数$f(x) ≡ 0$。

$\lim\limits_{x⇝x_0} g(f(x)) = g(0) = 0 ≠ 1 ⇜ \lim\limits_{y⇝0} g(y)$

函数$f (x)$在点$x_0$处可导,若点$x_0$为$f (x)$的极值点,则${^1}f (x_0) = 0$,反之不对。

函数$f (x)$在点$x_0$处可导,若${^1}f (x_0) = 0$,则点$x_0$为$f (x)$的停驻点,反之亦然。

反例:函数$f (x) = x $在点$x_0 = 0$处不可导。

若函数$f (x)$在区间$X$上严格单调,则反函数$’f (y)$在区间$Y$上的单调性相同。

$∀x_1,x_2∈X; [ {‘}f (y_1) = x_1 < x_2 = {‘}f (y_2) ] ⇔ [ f^{↑} (x_1) = y_1 < y_2 = f^{↑} (x_2) ]$

$∀x_1,x_2∈X; [ {‘}f (y_1) = x_1 < x_2 = {‘}f (y_2) ] ⇔ [ f^{↓} (x_1) = y_1 > y_2 = f^{↓} (x_2) ]$

$⇓$ $∀x_1,x_2∈X; [ x_1 < x_2 ] ⇒ [ f^{↑} (x_1) = y_1 < y_2 = f^{↑} (x_2) ]$ $P ⇒ Q$
$⇓$ $∀x_1,x_2∈X; [ x_1 = x_2 ] ⇒ [ f^{↑} (x_1) = y_1 = y_2 = f^{↑} (x_2) ]$ $R ⇒ S$
$⇓$ $∀x_1,x_2∈X; [ x_1 ≤ x_2 ] ⇒ [ f^{↑} (x_1) = y_1 ≤ y_2 = f^{↑} (x_2) ]$ $[P ∨ R] ⇒ [ Q ∨ S]$
$⇓$ $∀y_1,y_2∈X; [ y_1 > y_2 ] ⇒ [ {‘}f (y_1) = x_1 > x_2 = {‘}f (y_2) ]$ $¬[ Q ∨ S] ⇒ ¬[P ∨ R]$
$⇓$ $∀y_1,y_2∈X; [ y_1 < y_2 ] ⇒ [ {‘}f (y_1) = x_1 < x_2 = {‘}f (y_2) ]$  
     
$⇓$ $∀x_1,x_2∈X; [ {‘}f (y_1) = x_1 < x_2 = {‘}f (y_2) ] ⇔ [ f^{↑} (x_1) = y_1 < y_2 = f^{↑} (x_2) ]$  

若函数$f^{↕} (x)$在单区间$X$上严格单调,且其函数值域为单区间,则函数$f^{↕} (x)$在单区间$X$上单调连续。

$⇓$ $∀x_1,x_2∈X; [ f^{-1} (y_1) = x_1 < x_2 = f^{-1} (y_2) ] ⇔ [ f^{↑} (x_1) = y_1 < y_2 = f^{↑} (x_2) ]$                    
$⇓$ $∀y_1,y_0,y,y_2∈Y; [ y_1 < y_0, y < y_2 ] ⇔ [ x_1 < x_0, x < x_2 ]$                    
$⇓$ $∀ε= y_2-y_1 >0;∃δ= x_2-x_1 >0; [ x - x_0 < δ ] ⇒ [ y - y_0 < ε]$    
$⇓$ $∀ε>0;∃δ>0; [ x - x_0 < δ ] ⇒ [ f^{↑} (x) - f^{↑} (x_0) < ε ]$ $⇒$ $\lim\limits_{x⇝x_0} f^{↑} (x) ⇝ f^{↑} (x_0)$        
$⇓$     $\lim\limits_{x⇝x_0} f^{↕} (x) ⇝ f^{↕} (x_0)$                

连续性

函数$f(x)$在区间$X$上点$x_0$处振幅$w_{X}^{f}(x_0)$。若函数$f(x)$在区间$X$上点$x_0$处未定义,则其在该点处振幅$w_{X}^{f}(x_0)$可视为无穷大。

$w_{X}^{f}(x_0,δ) ≡ \sup\limits_{u,v∈\mathrm{B}_{X}(x_0,δ)} f(u) - f(v) = \left[ \sup\limits_{u∈\mathrm{B}{X}(x_0,δ)} f(u) - \inf\limits{v∈\mathrm{B}_{X}(x_0,δ)} f(v) \right]$
$w_{X}^{f}(x_0) ≡ \lim\limits_{δ⇝0} w_{X}^{f}(x_0,δ) ≡ \lim\limits_{δ⇝0} \sup\limits_{ u,v-x_0 ≤δ}^{u,v∈X} f(u) - f(v) = \lim\limits_{δ⇝0} \left[ \sup\limits_{ u-x_0 ≤δ}^{u∈X} f(u) - \inf\limits_{ v-x_0 ≤δ}^{v∈X} f(v) \right]$
$w^{f}(x_0) ≠ 0$ $∃ε>0;∀δ>0; \sup\limits_{u,v∈\mathrm{B}_{X}(x_0,δ)} f(u) - f(v) >ε$ $∃ε>0;∀δ>0;∃u,v∈\mathrm{B}_{X}(x_0,δ); f(u) - f(v) >ε$ $∃ε>0;∀δ>0;∃u,v∈X; u,v-x_0 ≤δ ∧ f(u) - f(v) >ε$
\(w^{f}(x_0) = 0\) $∀ε>0;∃δ>0; \sup\limits_{u,v∈\mathrm{B}_{X}(x_0,δ)} f(u) - f(v) ≤ ε$ $∀ε>0;∃δ>0;∀u,v∈\mathrm{B}_{X}(x_0,δ); f(u) - f(v) ≤ ε$ $∀ε>0;∃δ>0;∀u,v∈X; u,v-x_0 ≤δ ⇒ f(u) - f(v) ≤ ε$

连续性是局部点域性质。若函数$f(x)$在点$x_0∈X$处连续,则函数在点$x_0∈X$处既左连续也右连续,反之亦然。

$\left[ \lim\limits_{x⇝x_0} f (x) ⇝ f (x_0) \right] ⇔ \left[ \lim\limits_{x⇝x_0^{-} } f (x) ⇝ f (x_0) \right] ∧ \left[ \lim\limits_{x⇝x_0^{+} } f (x) ⇝ f (x_0) \right]$

$\lim\limits_{x⇝x_0^{+} } f(x) ⇝ f(x_0) ≠ ∞$ $Ⅎx_0∈X;∀ε>0;∃δ>0; \sup\limits_{x∈\mathrm{B}(x_0^{+},δ)} f(x) - f(x_0) ≤ε$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x∈\mathrm{B}(x_0^{+},δ); f(x)∈\mathrm{B}(f(x_0),ε)$        
$\lim\limits_{x⇝x_0^-} f(x) ⇝ f(x_0) ≠ ∞$ $Ⅎx_0∈X;∀ε>0;∃δ>0; \sup\limits_{x∈\mathrm{B}(x_0^{-},δ)} f(x) - f(x_0) ≤ε$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x∈\mathrm{B}(x_0^{-},δ); f(x)∈\mathrm{B}(f(x_0),ε)$        
$\lim\limits_{x⇝x_0} f (x) ⇝ f (x_0) ≠ ∞$ $Ⅎx_0∈X;∀ε>0;∃δ>0; \sup\limits_{x∈\mathrm{B}(x_0,δ)} f(x) - f(x_0) ≤ ε$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x∈\mathrm{B}(x_0,δ); f(x)∈\mathrm{B}(f(x_0),ε)$        
$\lim\limits_{x⇝x_0} f (x) ⇝ f (x_0) ≠ ∞$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x∈\mathrm{B}(x_0,δ); f(x) - f(x_0) ≤ε$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x∈X; x - x_0 ≤δ ⇒ f(x) - f(x_0) ≤ε$
                 
$\lim\limits_{x⇝x_0} f(x) \not⇝ f(x_0)$ $Ⅎx_0∈X;∃ε>0;∀δ>0;∃x∈\mathrm{B}(x_0,δ); f(x) - f(x_0) > ε$ $Ⅎx_0∈X;∃ε>0;∀δ>0;∃x∈X; x - x_0 ≤ δ ∧ f(x) - f(x_0) > ε$

一致连续性是全局区域性质。若函数$f(x)$在区间$X$上一致连续,则函数必在任意点$x∈X$处连续,反之不对。$[∃x;∀y; Q(x, y)] ⇒ [∀y;∃x; Q(x, y)]$

$\left[ \lim\limits_{x↭x_t} f (x) ↭ f (x_t) \right] ⇒ \left[ \lim\limits_{x⇝x_0} f(x) ⇝ f(x_0) \right]$

$\lim\limits_{x↭x_t} f (x) ↭ f (x_t) ≠ ∞$ $∀ε>0;∃δ>0;∀x,x_t∈X; x - x_t ≤δ ⇒ f(x) - f(x_t) ≤ε$ $∀ε>0;∃δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) ≤ε$
                     
$\lim\limits_{x↭x_t} f(x) \not↭ f(x_t)$ $∃ε>0;∀δ>0;∃x,x_t∈X; x - x_t ≤ δ ∧ f(x) - f(x_t) > ε$ $∃ε>0;∀δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε$

若函数$f (x)$在区间$X$上点$x_0$处连续,则在该点处振幅必为零,反之亦然。若函数$f(x)$在区间$X$上点$x_0$处间断,则在该点处振幅非为零,反之亦然。

$\left[ \lim\limits_{x⇝x_0} f (x) ⇝ f (x_0) \right] ⇔ \left[ \lim\limits_{δ⇝0} w^{f} (x_0,δ) ⇝ 0 \right]$

$\left[ \lim\limits_{x⇝x_0} f(x) \not⇝ f(x_0) \right] ⇔ \left[ \lim\limits_{δ⇝0} w^{f}(x_0,δ) \not⇝ 0 \right]$

$⇓$ $∀ε_u>0;∃δ_u>0;∀u∈\mathrm{B}(x_0,δ_u); f(u) - f(x_0) ≤ ε_u $ $\lim\limits_{x⇝x_0} f(x) ⇝ f(x_0)$        
$⇓$ $∀ε_u>0;∃δ_u>0;∀ε_v>0;∃δ_v>0;∀ε>0;∃δ>0;∀u,v∈\mathrm{B}(x_0,δ=\min\lbrace δ_u,δ_v \rbrace); f(u) - f(v) f(u) - f(x_0) + f(v) - f(x_0) ≤ ε_u + ε_v = ε$  
$⇓$ $∀ε>0;∃δ>0;∀u,v∈\mathrm{B}(x_0,δ); f(u) - f(v) ≤ ε$          
$⇓$ $∀ε>0;∃δ>0;∀x∈\mathrm{B}(x_0,δ); f(x) - f(x_0) ≤ ε$ $\lim\limits_{δ⇝0} w^{f} (x_0,δ) ⇝ 0$        

若函数$f(x)$在区间$X$上点$x_0$处左极限与右极限全都存在,但不均等于该点处函数值,则称点$x_0$为函数$f(x)$的跳跃间断点。若极限值为无穷大,则可视为极限存在。

$\left[ f(x) _{x_0^{-} } ≠ f(x_0) \right] ∨ \left[ f(x_0) ≠ f(x) _{x_0^{+} } \right]$

若函数$f(x)$在区间$X$上点$x_0$处左极限与右极限不都存在,但不均等于该点处函数值,则称点$x_0$为函数$f(x)$的振荡间断点。其极限值无限振荡,不趋于任何固定值。

$\left[ \lim\limits_{x⇝x_0^{-} } f(x) \not⇝ f(x_0) \right] ∨ \left[ f(x_0) \not⇜ \lim\limits_{x⇝x_0^{+} } f(x) \right]$

若函数$f (x)$在点$x_0$处连续$\lim\limits_{x⇝x_0} f (x) ⇝ f (x_0)$,且函数$g (x)$在点$x_0$处连续$\lim\limits_{x⇝x_0} g (x) ⇝ g (x_0)$。

$\lim\limits_{x⇝x_0} [ f (x) + g (x) ] = \lim\limits_{x⇝x_0} f (x) + \lim\limits_{x⇝x_0} g (x) ⇝ f (x_0) + g (x_0)$

$\lim\limits_{x⇝x_0} [ f (x) - g (x) ] = \lim\limits_{x⇝x_0} f (x) - \lim\limits_{x⇝x_0} g (x) ⇝ f (x_0) - g (x_0)$

$\lim\limits_{x⇝x_0} [ f (x) · g (x) ] = \lim\limits_{x⇝x_0} f (x) · \lim\limits_{x⇝x_0} g (x) ⇝ f (x_0) · g (x_0)$

$\lim\limits_{x⇝x_0} \dfrac{f (x)}{g (x)} = \dfrac{\lim\limits_{x⇝x_0} f (x)}{\lim\limits_{x⇝x_0} g (x)} \mathop{⇝}\limits_{g(x_0)≠0} \dfrac{f (x_0)}{g (x_0)}$

$\lim\limits_{x⇝x_0} \mathrm{Con} · f (x) = \mathrm{Con} · \lim\limits_{x⇝x_0} f (x) ⇝ \mathrm{Con} · f (x_0)$

若函数$f(x)$在点$x_0$处连续$\lim\limits_{x⇝x_0} f(x) ⇝ f(x_0) = y_0$,且函数$g(y)$在点$y_0$处连续$\lim\limits_{y⇝y_0} g(y) ⇝ g(y_0)$,则复合函数$g(f(x))$在点$x_0$处连续。

$\lim\limits_{x⇝x_0} g(f(x)) = g\left( \lim\limits_{x⇝x_0} f(x) \right) ⇝ g(f(x_0))$

$⇓$ $∀ζ>0;∃ε>0;∀y; y∈\mathrm{B}(y_0,ε) ⇒ g(y)∈\mathrm{B}(g(y_0),ζ)$  
$⇓$ $∀ε>0;∃δ>0;∀x; x∈\mathrm{B}(x_0,δ) ⇒ f(x)∈\mathrm{B}(y_0,ε)$  
$⇓$ $∀ζ>0;∃ε>0;∃δ>0;∀x; x∈\mathrm{B}(x_0,δ) ⇒ f(x)∈\mathrm{B}(y_0,ε) ⇒ g(f(x))∈\mathrm{B}(g(y_0),ζ)$  
$⇓$ $∀ζ>0;∃ε>0;∀y=f(x); y∈\mathrm{B}(x_0,ε) ⇒ g(y)∈\mathrm{B}(g(f(x_0)),ζ)$  
     
$⇓$ $\lim\limits_{x⇝x_0} g(f(x)) = \lim\limits_{y⇝y_0} g(y) = g(y_0) = g\left( \lim\limits_{x⇝x_0} f(x) \right) ⇝ g(f(x_0))$  

一致连续性

若函数$f (x)$在区间$\fbox{X}$上连续,则函数$f (x)$在区间$\fbox{X}$上一致连续,反之亦然。

$\left[ \lim\limits_{x⇝x_0} f (x) \mathop{⇝}\limits_{x,x_0∈\fbox{X} } f (x_0) \right] ⇔ \left[ \lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x,x_t∈\fbox{X} } f (x_t) \right]$

$⇓$ $∀x_0∈\fbox{X};∀ε>0;∃δ_0>0; w^{f}(x_0,δ_0) ≤ ε$ $⇔$ $∀x_0∈X;∀ε>0;∃δ_0>0;∀x∈X; x - x_0 ≤ δ_0 ⇒ f(x) - f(x_0) ≤ ε$
$⇓$ $∀ε>0;∀x_0∈\fbox{X};∃δ_0>0; w^{f}(x_0,δ_0) ≤ ε$            
$⇓$ $∀ε>0;∃δ>0;∀x_0∈\fbox{X};∃δ_0>0; w^{f}\left(x_0,δ=\inf\limits_{x_0∈\fbox{X} }δ_0 \right) ≤ w^{f}(x_0,δ_0) ≤ ε$            
$⇓$ $∀ε>0;∃δ>0;∀x∈\fbox{X}; w^{f}(x,δ) ≤ ε$ $⇔$ $∀ε>0;∃δ>0;∀x,x_t∈X; x - x_t ≤ δ ⇒ f(x) - f(x_t) ≤ ε$

若函数$f (x)$在区间$\fbox{X}$上导函数有确界,则函数$f (x)$在区间$X$上必一致连续,反之不对。$P ⇒ Q$

若函数$f(x)$在区间$X$上非一致连续,则函数$f(x)$在区间$\fbox{X}$上导函数无确界,反之不对。$¬Q ⇒ ¬P$

$\left[ ∀x_t∈\fbox{X};∃\mathrm{Sup}_{x}; \left \dfrac{\mathrm{d} f (x)}{\mathrm{d} x} \right ≤ \mathrm{Sup_{x} } \right] ⇒ \left[ ∃\mathrm{Sup}^{f};∀x,x_t∈X; f (x) - f (x_t) ≤ \mathrm{Sup}^{f} · x - x_t \right] ⇒ \left[ \lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x,x_t∈X} f (x_t) \right]$
$⇓$ $∀x∈\fbox{X};∃\mathrm{Sup}_{x}; \left \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right ≤ \mathrm{Sup}_{x} ≠ ∞^{+}$                
$⇓$ $∃\mathrm{Sup};∀x∈\fbox{X};∃\mathrm{Sup}_{x}; \left \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right ≤ \mathrm{Sup}{x} ≤ \mathrm{Sup} = \sup\limits{x∈\fbox{X} } \left\lbrace \mathrm{Sup}_{x} \right\rbrace$                
$⇓$ $∃\mathrm{Sup};∀x∈\fbox{X}; \mathrm{d} f(x) ≤ \mathrm{Sup} · \mathrm{d} x $ $⇔$ $∃\mathrm{Sup}^{f};∀x,x_t∈X; f(x) - f(x_t) ≤ \mathrm{Sup}^{f} · x - x_t $
$⇓$ $∃\mathrm{Sup};∀ε>0;∃δ>0;∀x,x_t∈X; [ x - x_t ≤ δ ] ⇒ [ f(x) - f(x_t) ≤ \mathrm{Sup} · x - x_t ≤ \mathrm{Sup} · δ = ε ]$        
$⇓$ $\lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x,x_t∈X} f (x_t)$                    

若函数$f (x)$在区间$(x_{α}, x_{β})$上一致连续,则可将函数$f(x)$延拓为区间$[x_{α}, x_{β}]$上的一致连续函数$\tilde{f}(x) = \mathop{f(x_α^+)}\limits_{x=x_α}; \mathop{f (x)}\limits_{x∈(x_α,x_β)};\mathop{f(x_β^{-})}\limits_{x=x_β}$。

若函数$f (x)$在区间$(x_α,x_β]$上一致连续,且在区间$[x_β, x_γ)$上一致连续,则函数$f (x)$在区间$(x_α, x_γ)$上一致连续。

若函数$f (x)$在区间$(∞^{-},∞^{+})$上连续,且以区间$T = [t_α, t_β]$为周期,则函数$f (x)$在区间$(∞^{-},∞^{+})$上一致连续。

$\left[ \lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈(x_α,x_β]} f(x_t) \right] ∧ \left[ \lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈[x_β,x_γ)} f(x_t) \right] ⇒ \left[ \lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈(x_α,x_γ)} f (x_t) \right]$

$⇓$ $∀ε_1>0;∃δ_1>0;∀x,x_t∈(x_α,x_β]; \left[ x - x_t ≤ δ_1 \right] ⇒ \left[ f (x) - f (x_t) ≤ ε_1 \right]$ $\lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x,x_t∈(x_α,x_β]} f (x_t)$        
$⇓$ $∀ε_2>0;∃δ_2>0;∀x,x_t∈[x_β,x_γ); \left[ x - x_t ≤ δ_2 \right] ⇒ \left[ f (x) - f (x_t) ≤ ε_2 \right]$ $\lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈[x_α,x_β)} f(x_t)$        
$⇓$ $∀ε_3>0;∃δ_3=\min\lbrace δ_1, δ_2 \rbrace;∀x∈(x_α,x_β];∀x_t∈[x_β,x_γ); [ x - x_t ≤ δ_3 ] ⇒ [ f(x) - f(x_t) f(x) - f(x_β) + f(x_β) - f(x_t) ≤ ε_1 + ε_2 = ε_3 ]$  
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈(x_α,x_γ); \left[ x - x_t ≤ δ \right] ⇒ \left[ f (x) - f (x_t) ≤ ε \right]$ $\lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈(x_α,x_β)} f(x_t)$        

若函数$f (x)$在区间$[x_α^{±}, ∞^{±})$上连续,且 $\lim\limits_{x⇝∞^{±} } f (x) ⇝ f_{∞^{±} }$极限存在,则函数$f (x)$在区间$[x_α^{±}, ∞^{±})$上一致连续。

$\left[ \lim\limits_{x⇝x_0} f (x) \mathop{⇝}\limits_{x,x_0∈[x_α^{±},∞^{±})} f (x_0) \right] ∧ \left[ \lim\limits_{x⇝∞^{±} } f (x) ⇝ f_{∞^{±} } \right] ⇔ \left[ \lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x,x_t∈[x_α^{±},∞^{±})} f (x_t) \right]$

$⇓$ $∃x_β;[x_α^{±}, ∞^{±}) = [x_α^{±}, x_β] ∪ [x_β, ∞^{±})$            
$⇓$ $∀ε>0;∃x_β∈[x_α^{±},∞^{±}];∀x,x_t∈[x_β,∞^{±}); [ f(x) - f(x_t) ≤ ε ]$ $⇔$ $\left[ \lim\limits_{x⇝∞^{±} } f (x) \mathop{⇝}\limits_{x∈[x_β,∞^{±})} f_{∞^{±} } \right]$    
$⇓$ $∀ε>0;∃x_β∈[x_α^{±},∞^{±}];∃δ>0;∀x,x_t∈[x_β,∞^{±}); ¬[ x - x_t ≤ δ ] ∨ [ f(x) - f (x_t) ≤ ε ]$    
$⇓$ $∀ε>0;∃x_β∈[x_α^{±},∞^{±}];∃δ>0;∀x,x_t∈[x_β,∞^{±}); [ x - x_t ≤ δ ] ⇒ [ f(x) - f(x_t) ≤ ε ]$ $⇒$ $\left[ \lim\limits_{x↭x_t} f (x) \mathop{↭}\limits_{x∈[x_β,∞^{±})} f (x_t) \right]$
$⇓$ $∀ε>0;∃x_β∈[x_α^{±},∞^{±}];∃δ>0;∀x,x_t∈[x_α^{±},x_β]; [ x - x_t ≤ δ ] ⇒ [ f(x) - f(x_t) ≤ ε ]$ $⇔$ $\left[ \lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈[x_α^{±},x_β]} f(x_t) \right]$
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈[x_α^{±},∞^{±}); [ x - x_t ≤ δ ] ⇒ [ f(x) - f(x_t) ≤ ε ]$ $⇒$ $\left[ \lim\limits_{x↭x_t} f(x) \mathop{↭}\limits_{x,x_t∈[x_α^{±},∞^{±})} f(x_t) \right]$

若函数$y = f(x)$在区间$X$上非一致连续,则其反函数$x = {‘}f(y)$在区间$Y$上一致连续,反之不对。

$\left[ ∃ε>0;∀δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε \right] ⇒ \left[ ∀δ>0;∃ε>0; \sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t ≤ δ \right]$
$⇓$ $∃ε>0;∀δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε$ $⇔$ $∃ε>0;∀δ>0;∃x,x_t∈X; x - x_t ≤ δ ∧ f(x) - f(x_t) > ε$
$⇓$ $∀δ>0; ∃ε>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε$   $∃x;∀y;P(x,y) ⇒ ∀y;∃x;P(x,y)$        
$⇓$ $∀δ>0;∃ε>0; \sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t ≤ δ$ $⇔$ $∀δ>0;∃ε>0;∀f(x),f(x_t)∈Y; f(x) - f(x_t) ≤ ε ⇒ x - x_t ≤ δ$
                       
$⇓$ $∃ε>0;∀δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) > ε$ $⇒$ $∀δ>0;∃ε>0; \sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t ≤ δ$
$⇓$ $∀ε>0;∃δ>0; \sup\limits_{ x-x_t ≤δ} f(x) - f(x_t) ≤ ε$ $⇐$ $∃δ>0;∀ε>0; \sup\limits_{ f(x)-f(x_t) ≤ε} x - x_t > δ$

典例:函数$f (x) = \dfrac{1}{x}$在区间$(0, x_β)$上非一致连续,但在区间$\mathop{[x_α, x_β)}\limits_{0<x_α}$上一致连续。

典例:函数$f (x) = α · x + β$在区间$(∞^{-}, ∞^{+})$上一致连续。函数$f (x) = \sin x^2$在区间$(∞^{-}, ∞^{+})$上非一致连续。

典例:函数$f (x) = x $在区间$(∞^{-}, ∞^{+})$上一致连续。函数$\mathrm{sgn} (x) = \mathop{-1}\limits_{x<0};\mathop{0}\limits_{x=0};\mathop{+1}\limits_{x>0}$在区间$[-1, +1]$上非一致连续。
典例:函数$f (x) = \sqrt{x}$在区间$[0, ∞^{+})$上一致连续,但在点$x_0=0^+$处导函数右极限为$\left.\dfrac{\mathrm{d} f (x)}{\mathrm{d} x} \right {0^+} = \lim\limits{x⇝0^+} \dfrac{1}{2 · \sqrt{x} } = ∞^{+}$。

典例:函数$f(x) = x^2$在区间$[0,∞^{+})$上非一致连续,函数$f(x) = x^2$与${‘}f(y) = \sqrt{y}$在区间$[0,∞^{+})$上互为反函数。

$⇓$ $∃ε>0;∀δ>0;∃x=\dfrac{1}{n},x_t=\dfrac{1}{n + 1}∈(0,x_β); \left[ x - x_t = \left \dfrac{1}{n · (n + 1)} \right ≤ δ \right] ∧ \left[ f (x) - f (x_t) = \left \dfrac{1}{x} - \dfrac{1}{x_t} \right = \left \dfrac{1}{\frac{1}{n} } - \dfrac{1}{\frac{1}{n + 1} } \right = 1 > ε \right]$    
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈[x_α,x_β); \left[ x - x_t ≤ δ \right] ⇒ \left[ f (x) - f (x_t) = \left \dfrac{1}{x} - \dfrac{1}{x_t} \right = \dfrac{ x - x_t }{ x · x_t } ≤ \dfrac{δ}{x_α^2} = ε \right]$    
                           
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈(∞^{-},∞^{+}); \left[ x - x_t ≤ δ \right] ⇒ \left[ f (x) - f (x_t) = \left (α · x + β) - (α · x_t + β) \right = α · x - x_t α · δ = ε \right]$
                           
$⇓$ $∃ε>0;∀δ>0;∃x=\sqrt{n·π+\frac{π}{2} },x_t=\sqrt{n·π}∈(∞^{-},∞^{+}); \left[ x - x_t = \dfrac{\frac{π}{2} }{\sqrt{n·π+\frac{π}{2} } + \sqrt{n·π} } ≤ δ \right] ∧ \left[ f (x) - f (x_t) = \left \sin x^2 - \sin x_t^2 \right = \left \sin \left( n·π+\frac{π}{2} \right) - \sin (n·π) \right = 1 > ε \right]$        
                           
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈(∞^{-},∞^{+}); [ x - x_t ≤ δ ] ⇒ [ f (x) - f (x_t) =   x - x_t   x - x_t ≤ δ = ε ]$
                           
$⇓$ $∃ε>0;∀δ>0;∃x=\dfrac{1}{n},x_t=0∈[-1,+1]; \left[ x - x_t = \left \dfrac{1}{n} \right ≤ δ \right] ∧ [ f (x) - f (x_t) = 1 - 0 = 1 > ε ]$        
                           
$⇓$ $∀ε>0;∃δ>0;∀x,x_t∈[0, ∞^{+}); [ x - x_t ≤ δ ] ⇒ \left[ f (x) - f (x_t) = \sqrt{x} - \sqrt{x_t} \mathop{≤}\limits^{\sqrt{u+v}≤\sqrt{u}+\sqrt{v} } \sqrt{ x - x_t } ≤ \sqrt{δ} = ε \right]$        
                           
$⇓$ $∃ε>0;∀δ>0;∃x=\sqrt{2·n·π+\frac{π}{2} },x_t=\sqrt{2·n·π}∈[0,∞^{+}); \left[ x - x_t = \dfrac{\frac{π}{2} }{\sqrt{2·n·π+\frac{π}{2} } + \sqrt{2·n·π} } ≤ δ \right] ∧ \left[ f(x) - f(x_t) = x^{2} - x_t^{2} = \dfrac{π}{2} > ε \right]$            

导函数

若函数$f(x)$在点$x_0$处去心闭邻域的右导数${^1}f(x_0^{+})$与左导数${^1}f(x_0^{-})$存在且相等,则其导数值${^1}f(x_0)$存在。函数$f(x)$在区间$X$上所有点$x_0$处导数值${^1}f(x_0)$组成其导函数${^1}f(x)$。

若函数$f(x)$在点$x_0$处导函数${^1}f(x)$的右极限${^1}f_{x_0^{+} }$与左极限${^1}f_{x_0^{-} }$存在且相等,则其导函数极限值${^1}f_{x_0}$存在。函数$f(x)$在点$x_0$处导数值${^1}f(x_0)$与其导函数极限值${^1}f_{x_0}$未必等同。

若函数$f(x)$在区间$(x_α,x_β)$上任意点处导数值存在,且在点$x_α$处右导数${^1}f(x_α^{+})$与在点$x_β$处左导数${^1}f(x_β^{-})$均存在,则可认为函数$f(x)$在区间$[x_α,x_β]$上任意点处导数值存在。

右导数 ${^1}f(x_0^+) ≡ \dfrac{\mathrm{d}^1 f(x_0^+)}{\mathrm{d}^1 x_0^{+} }$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x_t∈X; 0<x_t-x_0≤+δ ⇒ \left \dfrac{f(x_t) - f(x_0)}{x_t - x_0} - {^1}f(x_0) \right ≤ ε$      
左导数 ${^1}f(x_0^-) ≡ \dfrac{\mathrm{d}^1 f(x_0^-)}{\mathrm{d}^1 x_0^{-} }$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x_t∈X; -δ≤x_t-x_0<0 ⇒ \left \dfrac{f(x_t) - f(x_0)}{x_t - x_0} - {^1}f(x_0) \right ≤ ε$      
导数值 ${^1}f(x_0) ≡ \dfrac{\mathrm{d}^1 f(x_0)}{\mathrm{d}^1 x_0}$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t-x_0 ≤δ ⇒ \left \dfrac{f(x_t) - f(x_0)}{x_t - x_0} - {^1}f(x_0) \right ≤ ε$  
右导数 ${^1}f(x_0^+) ≡ \dfrac{\mathrm{d}^1 f(x_0^+)}{\mathrm{d}^1 x_0^{+} }$ $\lim\limits_{x_t⇝x_0^+} \dfrac{f(x_t) - f(x_0)}{x_t - x_0} \mathop{=====}\limits^{x_t=x_0+Δx_0} \lim\limits_{Δx_0⇝0^+} \dfrac{f(x_0+Δx_0) - f(x_0)}{Δx_0} = \lim\limits_{Δx_0⇝0^+} \dfrac{Δf(x_0)}{Δx_0}$          
左导数 ${^1}f(x_0^-) ≡ \dfrac{\mathrm{d}^1 f(x_0^-)}{\mathrm{d}^1 x_0^{-} }$ $\lim\limits_{x_t⇝x_0^-} \dfrac{f(x_t) - f(x_0)}{x_t - x_0} \mathop{=====}\limits^{x_t=x_0+Δx_0} \lim\limits_{Δx_0⇝0^-} \dfrac{f(x_0+Δx_0) - f(x_0)}{Δx_0} = \lim\limits_{Δx_0⇝0^-} \dfrac{Δf(x_0)}{Δx_0}$          
导数值 ${^1}f(x_0) ≡ \dfrac{\mathrm{d}^1 f(x_0)}{\mathrm{d}^1 x_0}$ $\lim\limits_{x_t⇝x_0} \dfrac{f(x_t) - f(x_0)}{x_t - x_0} \mathop{=====}\limits^{x_t=x_0+Δx_0} \lim\limits_{Δx_0⇝0} \dfrac{f(x_0 + Δx_0) - f(x_0)}{Δx_0} = \lim\limits_{Δx⇝0} \dfrac{Δf(x_0)}{Δx_0}$ ${^1}f(x_0^{-}) = {^1}f(x_0) = {^1}f(x_0^{+})$        
               
导函数 ${^1}f(x) ≡ \dfrac{\mathrm{d}^1 f(x)}{\mathrm{d}^1 x}$ $∀x∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t - x ≤ δ ⇒ \left \dfrac{f(x_t) - f(x)}{x_t - x} - {^1}f(x) \right ≤ ε$  
导函数 ${^1}f(x) ≡ \dfrac{\mathrm{d}^1 f(x)}{\mathrm{d}^1 x}$ $\lim\limits_{x_t⇝x} \dfrac{f(x_t) - f(x)}{x_t - x} \mathop{=====}\limits^{x_t=x+Δx} \lim\limits_{Δx⇝0} \dfrac{f(x + Δx) - f(x)}{Δx} = \lim\limits_{Δx⇝0} \dfrac{Δf(x)}{Δx}$          
导函数右极限 ${^1}f_{x_0^{+} } ≡ \left.\dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}\right _{x_0^{+} }$ $\lim\limits_{x⇝x_0^{+} } {^1}f(x) = \lim\limits_{x⇝x_0^{+} } \dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}$        
导函数左极限 ${^1}f_{x_0^{-} } ≡ \left.\dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}\right _{x_0^{-} }$ $\lim\limits_{x⇝x_0^{-} } {^1}f(x) = \lim\limits_{x⇝x_0^{-} } \dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}$        
导函数极限值 ${^1}f_{x_0} ≡ \left.\dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}\right _{x_0}$ $\lim\limits_{x⇝x_0} {^1}f(x) = \lim\limits_{x⇝x_0} \dfrac{\mathrm{d}^{1}f(x)}{\mathrm{d}^{1}x}$ ${^1}f_{x_0^{-} } = {^1}f_{x_0} = {^1}f_{x_0^{+} }$      
               
二阶导数值 ${^2}f(x_0) ≡ \dfrac{\mathrm{d}^{2} f(x_0)}{\mathrm{d}^{2} x_0}$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t - x_0 ≤ δ ⇒ \left \dfrac{ {^1}f(x_t) - {^1}f(x_0)}{x_t - x_0} - {^2}f(x_0) \right ≤ ε$  
二阶导数值 ${^2}f(x_0) ≡ \dfrac{\mathrm{d}^{2} f(x_0)}{\mathrm{d}^{2} x_0}$ $\lim\limits_{x_t⇝x_0} \dfrac{ {^1}f(x_t) - {^1}f(x_0)}{x_t - x_0} = \lim\limits_{Δx_0⇝0} \dfrac{ {^1}f(x_0 + Δx_0) - {^1}f(x_0)}{Δx_0} = \lim\limits_{Δx_0⇝0} \dfrac{Δ{^1}f(x_0)}{Δx_0}$ $\dfrac{\mathrm{d}^{2} f(x_0)}{\mathrm{d}^{2} x_0} ≡ \dfrac{\mathrm{d} }{\mathrm{d} x_0} \dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0}$        
二阶导函数 ${^2}f(x) ≡ \dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x}$ $∀x∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t - x ≤ δ ⇒ \left \dfrac{ {^1}f(x_t) - {^1}f(x)}{x_t - x} - {^2}f(x) \right ≤ ε$  
二阶导函数 ${^2}f(x) ≡ \dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x}$ $\lim\limits_{x_t⇝x} \dfrac{ {^1}f(x_t) - {^1}f(x)}{x_t - x} = \lim\limits_{Δx⇝0} \dfrac{ {^1}f(x + Δx) - {^1}f(x)}{Δx} = \lim\limits_{Δx⇝0} \dfrac{Δ{^1}f(x)}{Δx}$ $\dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} ≡ \dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$        
               
高阶导数值 ${^n}f(x_0) ≡ \dfrac{\mathrm{d}^{n} f(x_0)}{\mathrm{d}^{n} x_0}$ $Ⅎx_0∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t - x_0 ≤ δ ⇒ \left \dfrac{ {^{n-1} }f(x_t) - {^{n-1} }f(x_0)}{x_t - x_0} - {^{n} }f(x_0) \right ≤ ε$  
高阶导数值 ${^n}f(x_0) ≡ \dfrac{\mathrm{d}^{n} f(x_0)}{\mathrm{d}^{n} x_0}$ $\lim\limits_{x_t⇝x_0} \dfrac{ {^{n-1} }f(x_t) - {^{n-1} }f(x_0)}{x_t - x_0} = \lim\limits_{Δx_0⇝0} \dfrac{ {^{n-1} }f(x_0 + Δx_0) - {^{n-1} }f(x_0)}{Δx_0} = \lim\limits_{Δx_0⇝0} \dfrac{Δ{^1}f(x_0)}{Δx_0}$ $\dfrac{\mathrm{d}^{n} f(x_0)}{\mathrm{d}^{n} x_0} ≡ \dfrac{\mathrm{d} }{\mathrm{d} x_0} \dfrac{\mathrm{d}^{n-1} f(x_0)}{\mathrm{d}^{n-1} x_0}$        
高阶导函数 ${^n}f(x) ≡ \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x}$ $∀x∈X;∀ε>0;∃δ>0;∀x_t∈X; 0< x_t - x ≤ δ ⇒ \left \dfrac{ {^{n-1} }f(x_t) - {^{n-1} }f(x)}{x_t - x} - {^{n} }f(x) \right ≤ ε$  
高阶导函数 ${^n}f(x) ≡ \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x}$ $\lim\limits_{x_t⇝x} \dfrac{ {^{n-1} }f(x_t) - {^{n-1} }f(x)}{x_t - x} = \lim\limits_{Δx⇝0} \dfrac{ {^{n-1} }f(x + Δx) - {^{n-1} }f(x)}{Δx} = \lim\limits_{Δx⇝0} \dfrac{Δ{^1}f(x)}{Δx}$ $\dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x} ≡ \dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d}^{n-1} f(x)}{\mathrm{d}^{n-1} x}$        

典例:函数$f(x) = x^2$。

${^1}f(x) = \dfrac{\mathrm{d}^{1} f(x)}{\mathrm{d}^{1} x} = \lim\limits_{x_t⇝x} \dfrac{x_t^2 - x^2}{x_t - x} = \lim\limits_{x_t⇝x} (x_t + x) ⇝ 2 · x$

${^2}f(x) = \dfrac{\mathrm{d}^{2} f(x)}{\mathrm{d}^{2} x} = \lim\limits_{x_t⇝x} \dfrac{2 · x_t - 2 · x}{x_t - x} = \lim\limits_{x_t⇝x} 2 ⇝ 2$

$\lim\limits_{x_t⇝x} \dfrac{f(x_t) - f(x)}{(x_t - x)^2} = \lim\limits_{x_t⇝x} \dfrac{x_t^2 - x^2}{(x_t - x)^2} = \lim\limits_{x_t⇝x} \dfrac{x_t + x}{x_t - x} ⇝ ∞$

若函数$f(x)$在点$x_0$处一阶导数值${^1}f(x_0)$存在,则函数$f(x)$必定在点$x_0$处连续,反之不对。

若函数$f(x)$在点$x_0$处二阶导数值$^{2}f(x_0)$存在,则其一阶导函数在点$x_0$处连续,反之不对。

$\lim\limits_{x⇝x_0} f(x) = \lim\limits_{x⇝x_0} \left[ \dfrac{f(x) - f(x_0)}{x - x_0} · (x - x_0) + f(x_0) \right] = {^1}f(x_0) · \lim\limits_{x⇝x_0} (x - x_0) + f(x_0) ⇝ f(x_0)$

反例:函数$f(x) = x $在点$x_0 = 0$处连续,但其在点$x_0 = 0$处导数值不存在。

$\lim\limits_{x⇝0^{+} } \dfrac{f(x) - f(0)}{x - 0} = \lim\limits_{x⇝0^{+} } \dfrac{x}{x} ⇝ +1 ≠ -1 ⇜ \lim\limits_{x⇝0^{-} } \dfrac{-x}{x} = \lim\limits_{x⇝0^{-} } \dfrac{f(x) - f(0)}{x - 0}$

一阶导函数的运算性质。若函数$g(x)$与函数$f(x)$的一阶导函数存在。

$\dfrac{\mathrm{d} }{\mathrm{d} x}[g(x) + f(x)]$ $\dfrac{\mathrm{d} g(x)}{\mathrm{d} x} + \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$ $\lim\limits_{x_t⇝x} \dfrac{[g(x_t) + f(x_t)] - [g(x) + f(x)]}{x_t - x} = \lim\limits_{x⇝x} \dfrac{[g(x_t) - g(x)] + [f(x_t) - f(x)]}{x_t - x}$                                
$\dfrac{\mathrm{d} }{\mathrm{d} x}[g(x) - f(x)]$ $\dfrac{\mathrm{d}g(x)}{\mathrm{d}x} - \dfrac{\mathrm{d}f(x)}{\mathrm{d}x}$ $\lim\limits_{x_t⇝x} \dfrac{[g(x_t) - f(x_t)] - [g(x) - f(x)]}{x_t - x} = \lim\limits_{x_t⇝x} \dfrac{[g(x_t) - g(x)] - [f(x_t) - f(x)]}{x_t - x}$                                
$\dfrac{\mathrm{d} }{\mathrm{d}x}[g(x) · f(x)]$ $\dfrac{\mathrm{d} g(x)}{\mathrm{d} x} · f(x) + g(x) · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$ $\lim\limits_{x_t⇝x} \dfrac{g(x_t) · f(x_t) - g(x) · f(x)}{x_t - x} = \lim\limits_{x_t⇝x} \dfrac{[g(x_t) - g(x)] · f(x_t) + g(x) · [f(x_t) - f(x)]}{x_t - x}$                                
$\dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{g(x)}{f(x)}$ $\dfrac{\dfrac{\mathrm{d} g(x)}{\mathrm{d} x} · f(x) - g(x) · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} }{f^2(x)}$ $\lim\limits_{x_t⇝x} \dfrac{\dfrac{g(x_t)}{f(x_t)} - \dfrac{g(x)}{f(x)} }{x_t - x} \mathop{===}\limits^{f(x)≠0} \lim\limits_{x_t⇝x} \dfrac{\dfrac{[g(x_t) - g(x)] · f(x)}{x_t - x} - \dfrac{g(x) · [f(x_t) - f(x)]}{x_t - x} }{f(x_t) · f(x)}$                                
$\dfrac{\mathrm{d} }{\mathrm{d} x} g(f(x))$ $\dfrac{\mathrm{d} g(f(x))}{\mathrm{d} f(x)} · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \mathop{===}\limits^{y=f(x)} \dfrac{\mathrm{d} g(y)}{\mathrm{d} y} · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$ $\lim\limits_{x_t⇝x} \dfrac{g(f(x_t)) - g(f(x))}{x_t - x} = \lim\limits_{x_t⇝x} \dfrac{g(f(x_t)) - g(f(x))}{f(x_t) - f(x)} · \dfrac{f(x_t) - f(x)}{x_t - x}$                                
$\dfrac{\mathrm{d} }{\mathrm{d} x} {‘}f^{⇵}(x)$ $\left[ \dfrac{\mathrm{d} x}{\mathrm{d} {‘}f^{⇵}(x)} \right]^{-1} \mathop{====}\limits_{y={‘}f^{⇵}(x)}^{x=f^{⇵}(y)} \left[ \dfrac{\mathrm{d} f^{⇵}(y)}{\mathrm{d} y} \right]^{-1}$ $\lim\limits_{x_t⇝x} \dfrac{ {‘}f^{⇵}(x_t) - {‘}f^{⇵}(x)}{x_t - x} = \left[ \lim\limits_{x_t⇝x} \dfrac{x_t - x}{ {‘}f^{⇵}(x_t) - {‘}f^{⇵}(x)} \right]^{-1} \mathop{====}\limits_{y={‘}f^{⇵}(x)}^{x=f^{⇵}(y)} \left[ \lim\limits_{x_t⇝x} \dfrac{f^{⇵}(y_t) - f^{⇵}(y)}{y_t - y} \right]^{-1}$                                
                                     
$\dfrac{\mathrm{d} }{\mathrm{d} x} x $ $\dfrac{ x }{x}$ $\lim\limits_{x_t⇝x>0} \dfrac{(+x_t) - (+x)}{x_t - x} ⇝ +1;\lim\limits_{x_t⇝x<0} \dfrac{(-x_t) - (-x)}{x_t - x} ⇝ -1$                        
$\dfrac{\mathrm{d} }{\mathrm{d} x} f(x)^{g(x)}$ $f(x)^{g(x)} · \left[ \dfrac{\mathrm{d} g(x)}{\mathrm{d} x} · \ln f(x) + \dfrac{g(x)}{f(x)} · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right]$ $\dfrac{\mathrm{d} f(x)^{g(x)} }{\mathrm{d} f(x)^{g(x)} } · \dfrac{\mathrm{d} ә^{g(x) · \ln f(x) } }{\mathrm{d} x} = \dfrac{f(x)^{g(x)} }{ f(x)^{g(x)} } · ә^{g(x) · \ln f(x) } · \left[ \dfrac{\mathrm{d} g(x)}{\mathrm{d} x} · \ln f(x) + \dfrac{g(x)}{ f(x) } · \dfrac{ f(x) }{f(x)} · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right]$
典例:函数$f(x) = \dfrac{(x + 9)^{\frac{1}{2} } · (3 - x)^{\frac{2}{3} } }{(x + 4)^{\frac{3}{4} } · (5 - x)^{\frac{4}{5} } }$,因此$\ln f(x) = \dfrac{1}{2} · \ln x + 9 + \dfrac{2}{3} · \ln 3 - x - \dfrac{3}{4} · \ln x + 4 - \dfrac{4}{5} · \ln 5 - x $。

$\dfrac{1}{f(x)} · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{+1}{2 · (x + 9)} + \dfrac{-2}{3 · (3 - x)} - \dfrac{3}{4 · (x + 4)} - \dfrac{-4}{5 · (5 - x)} $

$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{(x + 9)^{\frac{1}{2} } · (3 - x)^{\frac{2}{3} } }{(x + 4)^{\frac{3}{4} } · (5 - x)^{\frac{4}{5} } } · \left[ \dfrac{+1}{2 · (x + 9)} + \dfrac{-2}{3 · (3 - x)} - \dfrac{3}{4 · (x + 4)} - \dfrac{-4}{5 · (5 - x)} \right]$

周期函数$f(x) = f(x + T)$,其导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} [f(x + T)]}{\mathrm{d} (x + T)} · \dfrac{\mathrm{d} (x + T)}{\mathrm{d} x} = \dfrac{\mathrm{d} f(x + T)}{\mathrm{d} (x + T)}$以$T$为周期。

奇函数$f(x) = -f(-x)$,其导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} [-f(-x)]}{\mathrm{d} (-x)} · \dfrac{\mathrm{d} (-x)}{\mathrm{d} x} = +\dfrac{\mathrm{d} f(-x)}{\mathrm{d} (-x)}$为偶函数。

偶函数$f(x) = +f(-x)$,其导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} [+f(-x)]}{\mathrm{d} (-x)} · \dfrac{\mathrm{d} (-x)}{\mathrm{d} x} = -\dfrac{\mathrm{d} f(-x)}{\mathrm{d} (-x)}$为奇函数。

高阶导函数的运算性质。若函数$g(x)$与函数$f(x)$的高阶导函数均存在。

$\dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) + f(x)]$ $\dfrac{\mathrm{d}^{n} g(x)}{\mathrm{d}^{n} x} + \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x}$ $\dfrac{\mathrm{d}^{0} f(x)}{\mathrm{d}^{0} x} ≡ f(x)$
$\dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) · f(x)]$ $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · \dfrac{\mathrm{d}^{n-i} g(x)}{\mathrm{d}^{n-i} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x}$  
$\dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [\mathrm{Con} · f(x)]$ $= \mathrm{Con} · \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x}$  
$\dfrac{\mathrm{d}^{0} }{\mathrm{d}^{0} x} [g(x) + f(x)]$ $g(x) + f(x)$ $\dfrac{\mathrm{d}^{0} f(x)}{\mathrm{d}^{0} x} ≡ f(x)$
$\dfrac{\mathrm{d}^{1} }{\mathrm{d}^{1} x} [g(x) + f(x)]$ $\dfrac{\mathrm{d}^{1} g(x)}{\mathrm{d}^{1} x} + \dfrac{\mathrm{d}^{1} f(x)}{\mathrm{d}^{1} x}$  
$\dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) + f(x)]$ $\dfrac{\mathrm{d}^{n} g(x)}{\mathrm{d}^{n} x} + \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x}$  
$\dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) + f(x)]$ $\dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \dfrac{\mathrm{d}^{n} g(x)}{\mathrm{d}^{n} x} + \dfrac{\mathrm{d}^{n} f(x)}{\mathrm{d}^{n} x} \right]$  
$\dfrac{\mathrm{d}^{n+1} }{\mathrm{d}^{n+1} x} [g(x) + f(x)]$ $\dfrac{\mathrm{d}^{n+1} g(x)}{\mathrm{d}^{n+1} x} + \dfrac{\mathrm{d}^{n+1} f(x)}{\mathrm{d}^{n+1} x}$  
     
$\dfrac{\mathrm{d}^{0} }{\mathrm{d}^{0} x} [g(x) · f(x)]$ $g(x) · f(x)$ $\dfrac{\mathrm{d}^{0} f(x)}{\mathrm{d}^{0} x} ≡ f(x)$
$\dfrac{\mathrm{d}^{1} }{\mathrm{d}^{1} x} [g(x) · f(x)]$ $\dfrac{\mathrm{d}^{1} g(x)}{\mathrm{d}^{1} x} · f(x) + g(x) · \dfrac{\mathrm{d}^{1} f(x)}{\mathrm{d}^{1} x}$  
$\dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) · f(x)]$ $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · \dfrac{\mathrm{d}^{n-i} g(x)}{\mathrm{d}^{n-i} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x}$  
$\dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d}^{n} }{\mathrm{d}^{n} x} [g(x) · f(x)]$ $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · \dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \dfrac{\mathrm{d}^{n-i} g(x)}{\mathrm{d}^{n-i} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x} \right]$  
  $\sum\limits_{i=0}^{n} \dfrac{n!}{i! · (n - i)!} · \left[ \dfrac{\mathrm{d}^{n-i+1} g(x)}{\mathrm{d}^{n-i+1} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x} + \dfrac{\mathrm{d}^{n-i} g(x)}{\mathrm{d}^{n-i} x} · \dfrac{\mathrm{d}^{i+1} f(x)}{\mathrm{d}^{i+1} x} \right]$  
  $\dfrac{\mathrm{d}^{n+1} g(x)}{\mathrm{d}^{n+1} x} · \dfrac{\mathrm{d}^{0} f(x)}{\mathrm{d}^{0} x} + \left[ \sum\limits_{i=1}^{n} \dfrac{n!}{i! · (n - i)!} · \dfrac{\mathrm{d}^{n-i+1} g(x)}{\mathrm{d}^{n-i+1} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x} + \sum\limits_{i=1}^{n} \dfrac{n!}{(i - 1)! · (n - i + 1)!} · \dfrac{\mathrm{d}^{n-(i-1)} g(x)}{\mathrm{d}^{n-(i-1)} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x} \right] + \dfrac{\mathrm{d}^{0} g(x)}{\mathrm{d}^{0} x} · \dfrac{\mathrm{d}^{n+1} f(x)}{\mathrm{d}^{n+1} x}$  
  $\dfrac{\mathrm{d}^{n+i} g(x)}{\mathrm{d}^{n+1} x} · \dfrac{\mathrm{d}^{0} f(x)}{\mathrm{d}^{0} x} + \left[ \sum\limits_{i=1}^{n} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · \dfrac{\mathrm{d}^{n+1-i} g(x)}{\mathrm{d}^{n+1-i} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x}\right] + \dfrac{\mathrm{d}^{0} g(x)}{\mathrm{d}^{0} x} · \dfrac{\mathrm{d}^{n+1} f(x)}{\mathrm{d}^{n+1} x}$  
$\dfrac{\mathrm{d}^{n+1} }{\mathrm{d}^{n+1} x} [g(x) · f(x)]$ $\sum\limits_{i=0}^{n+1} \dfrac{(n + 1)!}{i! · (n + 1 - i)!} · \dfrac{\mathrm{d}^{n+1-i} g(x)}{\mathrm{d}^{n+1-i} x} · \dfrac{\mathrm{d}^{i} f(x)}{\mathrm{d}^{i} x}$  

若函数$f(x)$在区间$X$上连续且导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ≶ 0$,则函数$f^{⇵}(x)$在区间$X$上严格单调,反之不对。

若函数$f(x)$在区间$X$上连续且导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ⪋ 0$,则函数$f^{⇵}(x)$在区间$X$上单调递进,反之亦然。

若函数$f(x)$在区间$X$上连续且严格单调,则其导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$仅有可能在单点处为零。

$\left[ \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ≶ 0 \right] ⇒ \left[ f^{⇵}(x) \right] ⇔ \left[ ∀x_1,x_2∈L≡\left\lbrace ∀x : \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = 0 \right\rbrace; [x_1,x_2]⊈L \right]$

$\left[ \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ⪋ 0 \right] ⇔ \left[ f^{⤨}(x) \right]$

$\lim\limits_{x_2⇝x_1} \dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \mathop{≤}\limits_{x_1=x_2} 0$ $⇐$ $\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_1,x_2)} < 0$ $⇒$ $[x_1 < x_2] ⇒ [f(x_1) < f(x_2)]$ $⇔$ $f^{↑}(x)$
$\lim\limits_{x_2⇝x_1} \dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ≤ 0$ $⇔$ $\dfrac{f(x_2) - f(x_1)}{x_2 - x_1} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_1,x_2)} ≤ 0$ $⇔$ $[x_1 < x_2] ⇒ [f(x_1) ≤ f(x_2)]$ $⇔$ $f^{↗}(x)$

典例:函数$f^{↑}(x) = x^3$,在$ℝ$上严格单调递增,但仅仅在点$x_0 = 0$处导数值${^1}f^{↑}(0) = 0$。

若函数$f(x)$在点$x_0$处的两侧导数值异号,则点$x_0$为极小值点异或者极大值点,反之不对。

若函数$f(x)$在点$x_0$处导数值为零,二阶导数值存在不为零,则点$x_0$为极值点,反之不对。

$\left[ {^1}f(x_0) = 0 \right] ∧ \left[ {^2}f(x_0) ≶ 0 \right] ⇒ \left[ {^1}f(x_0 - Δ) · {^1}f(x_0 + Δ) < 0 \right] ⇒ \left[ f(x_0) = \inf\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right] ∨ \left[ f(x_0) = \sup\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right]$

$⇓$ ${^1}f(x_0-Δ) · {^1}f(x_0+Δ) < 0$ $⇔$ $\dfrac{f(x_0-Δ) - f(x_0)}{-Δ} · \dfrac{f(x_0+Δ)-f(x_0)}{+Δ} < 0$
$⇓$ $\left[ f(x_0) = \inf\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right] ∨ \left[ f(x_0) = \sup\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right]$ $⇔$ $[f(x_0-Δx_0) - f(x_0)] · [f(x_0) - f(x_0+Δx_0)] < 0$
       
$⇓$ $\left[ {^1}f(x_0) = 0 \right] ∧ \left[ {^2}f(x_0) < 0 \right] ⇒ \left[ {^1}f(x_0 - Δ) · {^1}f(x_0 + Δ) < 0 \right]$   $f(x_0) = \sup\limits_{x∈\mathrm{B}(x_0,Δ)} f(x)$
$⇓$ $\left[ {^1}f(x_0) = 0 \right] ∧ \left[ {^2}f(x_0) > 0 \right] ⇒ \left[ {^1}f(x_0 - Δ) · {^1}f(x_0 + Δ) < 0 \right]$   $f(x_0) = \inf\limits_{x∈\mathrm{B}(x_0,Δ)} f(x)$
$⇓$ $\left[ {^1}f(x_0) = 0 \right] ∧ \left[ {^2}f(x_0) ≶ 0 \right] ⇒ \left[ {^1}f(x_0 - Δ) · {^1}f(x_0 + Δ) < 0 \right]$   $\left[ f(x_0) = \inf\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right] ∨ \left[ f(x_0) = \sup\limits_{x∈\mathrm{B}(x_0,Δ)} f(x) \right]$

若函数$f(x)$在区间$[x_α,x_β]$上连续,则函数$f(x)$在区间$[x_α,x_β]$上停驻点与曲折点处取得最小值与最大值。

$\inf\limits_{x∈[x_α,x_β]} f(x) = \inf \left\lbrace ∀x; f(x): x∈\left\lbrace ∀x_0∈[x_α,x_β]; x_0: \dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x} = 0 \right\rbrace \bigcup \left\lbrace ∀x_0∈[x_α,x_β]; x_0: \dfrac{\mathrm{d} f(x_0^{-})}{\mathrm{d} x} ≠ \dfrac{\mathrm{d} f(x_0^{+})}{\mathrm{d} x} \right\rbrace \right\rbrace$

$\sup\limits_{x∈[x_α,x_β]} f(x) = \sup \left\lbrace ∀x; f(x): x∈\left\lbrace ∀x_0∈[x_α,x_β]; x_0: \dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x} = 0 \right\rbrace \bigcup \left\lbrace ∀x_0∈[x_α,x_β]; x_0: \dfrac{\mathrm{d} f(x_0^{-})}{\mathrm{d} x} ≠ \dfrac{\mathrm{d} f(x_0^{+})}{\mathrm{d} x} \right\rbrace \right\rbrace$

典例:数列$n^{\frac{1}{n} }$在$3^{\frac{1}{3} }$处取得最大值。

$f(x) = x^{\frac{1}{x} }$ $\dfrac{\mathrm{d} f(x)}{\mathrm{d} x} = x^{\frac{1}{x} } · \left[ -\dfrac{1}{x^2} · \ln x + \dfrac{1}{x^2} · 1 \right] = x^{\frac{1}{x}-2} · (1 - \ln x ) = \mathop{+0}\limits_{x<ә};\mathop{0}\limits_{x=ә};\mathop{-0}\limits_{x>ә}$ $x^{\frac{1}{x}-2} > 0$
$\lim\limits_{x⇝0^{+} } x^{\frac{1}{x} } = \lim\limits_{x⇝0^{+} } ә^{\frac{1}{x}·\ln x} ⇝ ә^{∞^{-} } = 0$ $\lim\limits_{x⇝∞^{+} } x^{\frac{1}{x} } = \lim\limits_{x⇝∞^{+} } ә^{\frac{1}{x}·\ln x} ⇝ ә^{0^{+} } = 1$          
$\max\limits_{0≤n<∞^{+} } \left\lbrace n^{\frac{1}{n} } \right\rbrace = \max\limits_{2≤ә≤3} \left\lbrace 2^{\frac{1}{2} }, 3^{\frac{1}{3} } \right\rbrace = 3^{\frac{1}{3} }$            

连续介值定理

若函数$f (x)$在单区间$X$上连续,则其值域$Y$为单区间,反之不对。单区间内任意一点均为聚点,任意两点均可连通成线。

若函数$f(x)$在单区间$X=[x_{α},x_{β}]$上连续,则对于任意函数值$f(x)$,必定存在某一点$θ∈X$其函数值相等。

$f(x) _{∀x∈X} = \left.f (x)\right _{∃θ∈X}$
$⇓$ $∀x∈X;∃θ∈X; f (x) = f (θ)$ $⇔$ $\left. f(x) \right _{∀x∈X} = \left. f(x) \right _{∃θ∈X}$
$⇓$ $∀y∈f(X);∃θ∈X; y = f(θ)$ $⇐$ $y ≡ f(x)$    
$⇓$ $∀y∈[\min\lbrace f(x_α),f(x_β) \rbrace,\max\lbrace f(x_α),f(x_β) \rbrace] ⊆ f([x_α,x_β]);∃θ∈X; y = f(θ)$        
$⇓$ $[f(x_α) · f(x_β) ≤ 0] ⇒ \left[ 0 = \left. f(x) \right _{∃θ∈[x_α,x_β]} \right]$   $[f(x_α) · f(x_β) < 0] ⇒ \left[ 0 = \left. f(x) \right _{∃θ∈(x_α,x_β)} \right]$

若函数$f(x)$在单区间$X = [x_α, x_β]$上连续,对于权均值$\sum\limits_{i=0}^{n} t_i · f(x_i)$,则必定存在某一点$θ∈X$其函数值相等。

$\sum\limits_{i=0}^{n} t_i · f(x_i) \mathop{=====}\limits_{x_i∈[x_α,x_β]}^{\sum\limits_{i=0}^{n} t_i \mathop{==}\limits^{0≤t_i} 1} \left. f(θ) \right _{∃θ∈[x_α,x_β]}$
$⇓$ $\sum\limits_{i=0}^{n} t_i · f(x_i) ∈ [\min\lbrace f(x_i) \rbrace, \max\lbrace f(x_i) \rbrace] ⊆ f([x_α,x_β])$ $\sum\limits_{i=0}^{n} t_i \mathop{==}\limits^{0≤t_i} 1$  
$⇓$ $\sum\limits_{i=0}^{n} t_i · f(x_i) = \left. f(θ) \right _{∃θ∈[x_α,x_β]}$  

若函数$f (x)$在单区间$X = [x_α, x_β]$上连续,且其值域内含于定义域,则必定存在某一个不动点$θ∈[x_α, x_β]$。

$f([x_α,x_β]) ⊆ [x_α,x_β] ⇒ \left. f(x) \right _{∃θ∈[x_α,x_β]} = θ$
$⇓$ $F (x) ≡ f (x) - x$      
$⇓$ $F(x_α) · F(x_β) ≤ 0$ $⇐$ $F(x_α) = f(x_α) - x_α ≥ 0, F(x_β) = f(x_β) - x_β ≤ 0$  
$⇓$ $0 = \left. F (x) \right _{∃θ∈[x_α, x_β]}$    
$⇓$ $\left.f (x)\right _{∃θ∈[x_α, x_β]} = θ$    

对于实数域奇数次多项式$P_{2·n+1}(x) ≡ \sum\limits_{i=0}^{2·n+1} p_i · (x - x_0)^{i}$,必定存在某一个实数解$θ∈ℝ$。

$⇓$ $P_{2·n+1}(x) ≡ \sum\limits_{i=0}^{2·n+1} p_i · (x - x_0)^{i} = p_{2·n+1} · (x - x_0)^{2·n+1} · \left[ 1 + \sum\limits_{i=1}^{2·n} \dfrac{p_{i} · (x - x_0)^{i} }{p_{2·n+1} · (x - x_0)^{2·n+1} } + \dfrac{p_0}{p_{2·n+1}·(x - x_0)^{2·n+1} } \right]$      
$⇓$ $\lim\limits_{x⇝∞^{+} } P_{2·n+1}(x) = \lim\limits_{x⇝∞^{+} } p_{2·n+1} · (x - x_0)^{2·n+1} ⇝ p_{2·n+1} · ∞^{+}$      
$⇓$ $\lim\limits_{x⇝∞^{-} } P_{2·n+1}(x) = \lim\limits_{x⇝∞^{+} } p_{2·n+1} · (x - x_0)^{2·n+1} ⇝ p_{2·n+1} · ∞^{-}$      
$⇓$ $P_{2·n+1}(ℝ) = ℝ$ $⇒$ $0 ∈ P_{2·n+1}(ℝ)$  
$⇓$ $0 = \left. P_{2·n+1}(x) \right _{∃θ∈ℝ}$    

若函数$f (x)$在单区间$(x_α, x_β)$上连续,且$\lim\limits_{x⇝x_α^{+} } f (x)$与$\lim\limits_{x⇝x_β^{-} } f (x)$均有确界,则函数$f (x)$在区间$(x_α, x_β)$上有确界。

若函数$f (x)$在区间$(∞⁻, ∞⁺)$上连续,且$\lim\limits_{x⇝∞⁻} f (x)$与$\lim\limits_{x⇝∞⁺} f (x)$均有确界,则函数$f (x)$在区间$(∞⁻, ∞⁺)$上有确界。

$ f(x) ≤ \max\left\lbrace \left \inf\limits_{x∈X} f(x) \right , \left \sup\limits_{x∈X} f(x) \right \right\rbrace$

微分中值定理

若函数$f(x)$在点$x_0$处连续且导数值$\dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0} = 0$,则称点$x_0$为函数$f(x)$的停驻点。

若函数$f(x)$在点$x_0$处连续但其导数值并不存在,则称点$x_0$为函数$f(x)$的曲折点。

若函数$f(x)$在点$x_0$处达到极小值极大值且导数值存在,则其在点$x_0$处导数$\dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x} = 0$。

若函数$f(x)$在闭区间$[x_α,x_β]$上连续且导函数存在,当两端点处函数值相等,或者两端点处单调性相反,则存在某一点$θ∈(x_α,x_β)$其导数值为零。

$[f(x_α) = f(x_β)] ⇒ \left[ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = 0 \right]$
$[[f(x_α) - f(x_α+Δ)] · [f(x_β-Δ) - f(x_β)] ≤ 0] ⇒ \left[ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = 0 \right]$
$⇓$ $f(θ) = \inf\limits_{x∈(x_α,x_β)} f(x)$ $⇔$ $\lim\limits_{x⇝θ^{-} } \dfrac{f(x) - f(θ)}{x - θ} ≤ 0 ≤ \lim\limits_{x⇝θ^{+} } \dfrac{f(x) - f(θ)}{x - θ}$  
$⇓$ $f(θ) = \sup\limits_{x∈(x_α,x_β)} f(x)$ $⇔$ $\lim\limits_{x⇝θ^{-} } \dfrac{f(x) - f(θ)}{x - θ} ≥ 0 ≥ \lim\limits_{x⇝θ^{+} } \dfrac{f(x) - f(θ)}{x - θ}$  
$⇓$ $[f(θ-Δ) - f(θ)] · [f(θ) - f(θ+Δ)] ≤ 0$ $⇔$ $\dfrac{f(θ-Δ) - f(θ)}{-Δ} · \dfrac{f(θ+Δ) - f(θ)}{+Δ} ≤ 0$  
$⇓$ $f(x_γ) = f(x_δ)$ $⇒$ $\left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right {∃θ∈(x_γ,x_δ)} = \lim\limits{x⇝θ} \dfrac{f(x) - f(θ)}{x - θ} = 0$
         
$⇓$ $[[f(x_α) - f(x_α+Δ)] · [f(x_β-Δ) - f(x_β)] ≤ 0] ⇒ \left[ f(x_γ) \mathop{======}\limits_{x_α≤x_γ<x_δ≤x_β}^{∃x_γ,x_δ;} f(x_δ) \right]$ $⇒$ $\left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right {∃θ∈(x_γ,x_δ)⊆(x_α,x_β)} = \lim\limits{x⇝θ} \dfrac{f(x) - f(θ)}{x - θ} = 0$

若函数$f(x)$在闭区间$[x_α,x_β]$上连续且导函数存在,则存在某一点$θ∈(x_α,x_β)$其导数值,等同于两端点处直线的斜率。

$\dfrac{f(x_β) - f(x_α)}{x_β - x_α} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)}$
$⇓$ $F(x) ≡ \left \begin{matrix} 1 & 1 & 1 \ x_α & x & x_β \ f(x_α) & f(x) & f(x_β) \ \end{matrix}\right = \left \begin{matrix} 1 & 0 & 0 \ x_α & x - x_α & x_β - x_α \ f(x_α) & f(x) - f(x_α) & f(x_β) - f(x_α) \end{matrix}\right $ $F(x_α) = 0 = F(x_β)$
$⇓$ $\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} }{\mathrm{d} x} \left \begin{matrix} 1 & 0 & 0 \ x_α & x - x_α & x_β - x_α \ f(x_α) & f(x) - f(x_α) & f(x_β) - f(x_α) \ \end{matrix}\right = \left \begin{matrix} 1 & 0 & 0 \ x_α & 1 & x_β - x_α \ f(x_α) & \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} & f(x_β) - f(x_α) \end{matrix}\right $ $\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = [f(x_β) - f(x_α)] - (x_β - x_α) · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$
$⇓$ $\dfrac{f(x_β) - f(x_α)}{x_β - x_α} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)}$ $\left. \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = 0$    

若函数$f(x)$与函数$g(x)$在闭区间$[x_α,x_β]$上连续且导函数存在,则存在某一点$θ∈(x_α,x_β)$其导数值,等同于两端点处函数值的比值。

$\dfrac{g(x_β) - g(x_α)}{f(x_β) - f(x_α)} = \left.\dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)}\right _{∃θ∈(x_α, x_β)}$
$⇓$ $F(x) ≡ \left \begin{matrix} 1 & 1 & 1 \ g(x_α) & g(x) & g(x_β) \ f(x_α) & f(x) & f(x_β) \ \end{matrix}\right = \left \begin{matrix} 1 & 0 & 0 \ g(x_α) & g(x) - g(x_α) & g(x_β) - g(x_α) \ f(x_α) & f(x) - f(x_α) & f(x_β) - f(x_α) \ \end{matrix}\right $ $F(x_α) = 0 = F(x_β)$
$⇓$ $\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} }{\mathrm{d} x} \left \begin{matrix} 1 & 0 & 0 \ g(x_α) & g(x) - g(x_α) & g(x_β) - g(x_α) \ f(x_α) & f(x) - f(x_α) & f(x_β) - f(x_α) \ \end{matrix}\right = \left \begin{matrix} 1 & 0 & 0 \ g(x_α) & \dfrac{\mathrm{d} g(x)}{\mathrm{d} x} & g(x_β) - g(x_α) \ f(x_α) & \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} & f(x_β) - f(x_α) \end{matrix}\right $ $\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = [f(x_β) - f(x_α)] · \dfrac{\mathrm{d} g(x)}{\mathrm{d} x} - [g(x_β) - g(x_α)] · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$
$⇓$ $\dfrac{g(x_β) - g(x_α)}{f(x_β) - f(x_α)} = \left.\dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{∃θ∈(x_α,x_β)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} x} · \dfrac{\mathrm{d} x}{\mathrm{d} f(x)} \right _{∃θ∈(x_α,x_β)}$ $\left. \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = 0$  

导数介值定理

若函数$f(x)$在点$x_0$处连续,且其导函数的极限值$\left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0}$存在,则在该点处其导数值$\dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0}$等于该极限值,反之不对。若函数在无穷点处的函数值固定,则同样适用。
$\left[ \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} ⇝ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right {x_0} \right] ⇒ \left[ \lim\limits{x⇝x_0} \dfrac{f(x) - f(x_0)}{x - x_0} = \dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0} \right]$
$⇓$ $\dfrac{f(x) - f(x_0)}{x - x_0} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_0,x)}$ $⇐$ $\lim\limits_{x⇝x_0} f(x) ⇝ f(x_0)$        
$⇓$ $\dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0} = \lim\limits_{x⇝x_0} \dfrac{f(x) - f(x_0)}{x - x_0} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0}$ $⇐$ $\left[ \lim\limits_{x⇝x_0} \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∀x} ⇝ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right {x_0} \right] ⇒ \left[ \lim\limits{θ⇝x_0} \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ} ⇝ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0} \right]$

反例:函数$\mathcal{W}{2}(x) = \mathop{0}\limits{x=0};\mathop{x^{2} · \sin \dfrac{1}{x} }\limits_{x≠0}$,在点$x_0 = 0$处连续,但其导函数在点$x_0=0$处的极限值不存在。

$\dfrac{\mathrm{d} \mathcal{W}{2}(x)}{\mathrm{d} x} = \mathop{\left[ x · \sin \dfrac{1}{x} \right]}\limits{x=0};\mathop{\left[ 2 · x · \sin \dfrac{1}{x} - \cos \dfrac{1}{x} \right]}\limits_{x≠0}$

$\dfrac{\mathrm{d} \mathcal{W}{2}(0)}{\mathrm{d} 0} = \lim\limits{x⇝0} x · \sin \dfrac{1}{x} ⇝ 0 \not⇜ \lim\limits_{x⇝0} \dfrac{\mathrm{d} \mathcal{W}_{2}(x)}{\mathrm{d} x}$

若函数$g(x)$与函数$f(x)$在点$x_0$处的极限值均存在,且其导函数比值的极限值$\left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0}$存在,则在该点处其导函数比值的极限值$\dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)}$等于该极限值,反之不对。
$\left[ \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} ⇝ \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right {x_0} \right] ⇒ \left[ \lim\limits{x⇝x_0} \dfrac{g(x) - g(x) _{x_0} }{f(x) - f(x) _{x_0} } = \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0} \right]$
$\left[ \lim\limits_{x⇝x_0} \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} ⇝ \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right {x_0} \right] ⇒ \left[ \lim\limits{x⇝x_0} \dfrac{g(x)}{f(x)} \mathop{=====}\limits_{f(x) _{x_0} = 0}^{g(x) _{x_0} = 0} \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right {x_0} \right] ⇒ \left[ \lim\limits{x⇝x_0} \dfrac{g(x)}{f(x)} \mathop{=====}\limits_{f(x) _{x_0} = ∞}^{g(x) _{x_0} = ∞} \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0} \right]$
$⇓$ $\dfrac{g(x) - g(x) _{x_0} }{f(x) - f(x) _{x_0} } = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{∃θ∈(x_0,x)}$ $⇐$ $\lim\limits_{x⇝x_0} g(x) ⇝ g(x) _{x_0}$      
$⇓$ $\lim\limits_{x⇝x_0} \dfrac{g(x) - g(x) _{x_0} }{f(x) - f(x) _{x_0} } = \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0}$ $⇐$ $\left[ \lim\limits_{x⇝x_0} \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{∀x} ⇝ \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right {x_0} \right] ⇒ \left[ \lim\limits{x⇝x_0} \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{∃θ} ⇝ \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0} \right]$
                     
$⇓$ $\lim\limits_{x⇝x_0} \dfrac{g(x)}{f(x)} \mathop{=====}\limits_{f(x) _{x_0} = 0}^{g(x) _{x_0} = 0} \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0}$            
$⇓$ $\lim\limits_{x⇝x_0} \dfrac{g^{-1}(x)}{f^{-1}(x)} \mathop{=====}\limits_{f(x) _{x_0}=∞}^{g(x) {x_0}=∞} \left[ \lim\limits{x⇝x_0} \dfrac{(-1) · g^{-2}(x)}{(-1) · f^{-2}(x)} \right] · \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)}$              
$⇓$ $\lim\limits_{x⇝x_0} \dfrac{g(x)}{f(x)} = \lim\limits_{x⇝x_0} \dfrac{g^{-1}(x)}{f^{-1}(x)} · \lim\limits_{x⇝x_0} \dfrac{g^{2}(x)}{f^{2}(x)} = \dfrac{\mathrm{d} g(x_0)}{\mathrm{d} f(x_0)} = \left. \dfrac{\mathrm{d} g(x)}{\mathrm{d} f(x)} \right _{x_0}$                

反例:导数介值定理的前提条件不可忽略。

$\left[ \lim\limits_{x⇝∞^{+} } \dfrac{2 · x - \sin x}{2 · x + \cos x} = \lim\limits_{x⇝∞^{+} } \dfrac{2 - \dfrac{\sin x}{x} }{2 + \dfrac{\cos x}{x} } \right] ⇝ 1 \not⇜ \left[ \lim\limits_{x⇝∞^{+} } \dfrac{2 - \cos x}{2 - \sin x} = \lim\limits_{x⇝∞^{+} } \dfrac{2 · x - \sin x}{2 · x + \cos x} \right]$

典例:若函数$f(x)$在区间$X$上二阶导函数存在。

$\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} = \lim\limits_{h⇝0} \dfrac{f(x+h) + f(x-h) - 2 · f(x)}{h^2}$

$⇓$ $\lim\limits_{h⇝0} \dfrac{f(x+h) + f(x-h) - 2 · f(x)}{h^2}$
$⇓$ $= \lim\limits_{h⇝0} \dfrac{\dfrac{\mathrm{d} [f(x+h) + f(x-h) - 2 · f(x)]}{\mathrm{d} h} }{2 · h} = \lim\limits_{h⇝0} \dfrac{\dfrac{\mathrm{d} f(x+h)}{\mathrm{d} (x+h)} · \dfrac{\mathrm{d} (x+h)}{\mathrm{d} h} + \dfrac{\mathrm{d} f(x-h)}{\mathrm{d}(x-h)} · \dfrac{\mathrm{d} (x-h)}{\mathrm{d} h} }{2 · h}$
$⇓$ $= \lim\limits_{h⇝0} \dfrac{\dfrac{\mathrm{d} f(x+h)}{\mathrm{d} (x+h)} - \dfrac{\mathrm{d} f(x-h)}{\mathrm{d} (x-h)} }{2·h} = \lim\limits_{h⇝0} \dfrac{\dfrac{\mathrm{d} f(x+h)}{\mathrm{d}(x+h)} - \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} }{+2 · h} + \lim\limits_{h⇝0} \dfrac{\dfrac{\mathrm{d} f(x-h)}{\mathrm{d} (x-h)} - \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} }{-2 · h}$
$⇓$ $= \dfrac{\mathrm{d}^2 f(x)}{2 · \mathrm{d}^2 x} + \dfrac{\mathrm{d}^2 f(x)}{2 · \mathrm{d}^2 x} = \dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x}$

若函数$f(x)$在区间$[x_α,x_β]$上连续且导数值存在,则对于两端点处导数值之间任意取值$t$,存在某点$θ∈(x_α,x_β)$其导数值相等。

$∀t∈\left[\min\left\lbrace \dfrac{\mathrm{d} f(x_α^{+})}{\mathrm{d} x_α^{+} }, \dfrac{\mathrm{d} f(x_β^{-})}{\mathrm{d} x_{β}^{-} } \right\rbrace, \max\left\lbrace \dfrac{\mathrm{d} f(x_α^{+})}{\mathrm{d} x_α^{+} }, \dfrac{\mathrm{d} f(x_β^{-})}{\mathrm{d} x_β^{-} } \right\rbrace\right];∃θ∈(x_α,x_β); \dfrac{\mathrm{d} f(θ)}{\mathrm{d} x} = t$

$⇓$ $F(x) ≡ f(x) - t · x$   $\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} - t$      
$⇓$ $\dfrac{\mathrm{d} F(x_α^{+})}{\mathrm{d} x_α^{+} } · \dfrac{\mathrm{d} F(x_β^{-})}{\mathrm{d} x_{β^{-} } } = \left[ \dfrac{\mathrm{d} f(x_α^{+})}{\mathrm{d} x_{α}^{+} } - t \right] · \left[ \dfrac{\mathrm{d} f(x_β^{-})}{\mathrm{d} x_β^{-} } - t \right] ≤ 0$ $⇐$ $∀t∈\left[\min\left\lbrace \dfrac{\mathrm{d} f(x_α^{+})}{\mathrm{d} x_α^{+} }, \dfrac{\mathrm{d} f(x_β^{-})}{\mathrm{d} x_β^{-} } \right\rbrace, \max\left\lbrace \dfrac{\mathrm{d} f(x_α^{+})}{\mathrm{d} x_α^{+} }, \dfrac{\mathrm{d} f(x_β^{-})}{\mathrm{d} x_β^{-} } \right\rbrace\right];$      
$⇓$ $\dfrac{\mathrm{d} F(x_α^{+})}{\mathrm{d} x_α^{+} } = \lim\limits_{x⇝x_α^{+} } \dfrac{F(x) - F(x_α)}{x - x_α}$   $\dfrac{\mathrm{d} F(x_β^{-})}{\mathrm{d} x_β^{-} } = \lim\limits_{x⇝x_β^{-} } \dfrac{F(x) - F(x_β)}{x - x_β}$      
$⇓$ $[F(x_α) - F(x_α+Δ)] · [F(x_β-Δ) - F(x_β)] ≤ 0$   $\lim\limits_{x⇝x_0}^{x_0∈[x_α,x_β]} F(x) ⇝ F(x_0)$      
$⇓$ $∃θ∈(x_α,x_β); [F(θ-Δ) - F(θ)] · [F(θ) - F(θ+Δ)] ≤ 0$ $⇔$ $\left[ F(θ) = \inf\limits_{x∈\mathrm{B}(θ,Δ)} F(x) \right] ∨ \left[ F(θ) = \sup\limits_{x∈\mathrm{B}(θ,Δ)} F(x) \right]$      
$⇓$ $\left. \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} - t = 0$ $⇒$ $\left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{∃θ∈(x_α,x_β)} = t$

典例:不定式$\lim\limits_{x⇝∞^{+} } \left[ \sqrt[9]{x^9 + x^5} - \sqrt[9]{x^9 - x^5} \right]$的极限值。

$\lim\limits_{x⇝∞^{+} } x · \left[ (1 + \dfrac{1}{x^4})^{\frac{1}{9} } - (1-\dfrac{1}{x^{4} })^{\frac{1}{9} } \right] = \lim\limits_{t⇝0^{+} } \dfrac{(1+t^4)^{\frac{1}{9} } - (1-t^4)^{\frac{1}{9} } }{t} = \lim\limits_{t⇝0^{+} } \left[ \dfrac{1}{9} · (1+t^4)^{-\frac{8}{9} } · 4 · t^{3} + \dfrac{1}{9} · (1 - t^4)^{-\frac{8}{9} } · 4 · t^3 \right] ⇝ 0$

幂级数

多项式函数$\mathrm{Poly}_{n}(x)$在点$x_0$处连续,且有$n$导数值,在点$x_0$处必可唯一展开成$n$阶幂级数。

$\mathrm{P}{n}(x) = \sum\limits{i=0}^n p_i · (x - x_0)^i = p_0 · (x - x_0)^0 + p_1 · (x - x_0)^1 + p_2 · (x - x_0)^2 + ··· + p_n · (x - x_0)^n$

$\mathrm{P}{n}(x) = \sum\limits{i=0}^{n} \dfrac{ {^i}\mathrm{P}{n}(x_0)}{i!} · (x - x_0)^{i} = \sum\limits{i=0}^{n} \dfrac{\mathrm{d}^{i} \mathrm{P}_{n}(x_0)}{i! · \mathrm{d}^{i} x_0} · (x - x_0)^{i}$

  ${^i}\mathrm{P}{n} (x_0) = \left. {^i}\mathrm{P}{n}(x) \right _{x_0}$ $\left. {^i}\mathrm{P}_{n}(x) \right {x_0} = i! · p{i}$ $\left. {^0}\mathrm{P}_{n}(x) \right _{x_0} = 0! · p_0$ $\left. {^1}\mathrm{P}_{n}(x) \right _{x_0} = 1! · p_1$ $\left. {^2}\mathrm{P}_{n}(x_0) \right _{x_0} = 2! · p_2$
$\dfrac{\mathrm{d}^{i+1} \mathrm{P}_n(x)}{\mathrm{d}^{i+1} x} = \dfrac{\mathrm{d} }{\mathrm{d} x} \dfrac{\mathrm{d}^{i}\mathrm{P}_n(x)}{\mathrm{d}^{i} x}$ $\dfrac{\mathrm{d}^i \mathrm{P}{n}(x_0)}{\mathrm{d}^{i} x_0} = \left. \dfrac{\mathrm{d}^{i}\mathrm{P}{n}(x)}{\mathrm{d}^{i} x} \right _{x_0}$ $\left. \dfrac{\mathrm{d}^{i} \mathrm{P}_n (x)}{\mathrm{d}^{i} x} \right _{x_0} = i! · p_i$ $\left. \dfrac{\mathrm{d}^{0} \mathrm{P}_{n}(x)}{\mathrm{d}^{0} x} \right _{x_0} = 0! · p_0$ $\left. \dfrac{\mathrm{d}^{1} \mathrm{P}_{n}(x)}{\mathrm{d}^{1} x} \right _{x_0} = 1! · p_1$ $\left. \dfrac{\mathrm{d}^{2} \mathrm{P}_{n}(x)}{\mathrm{d}^2 x} \right _{x_0} = 2! · p_2$

任意的函数$f(x)$在点$x_0$处连续,若有$n+1$阶导数值则在点$x_0$处可唯一展开成$n$阶幂级数。

$f(x) = \sum\limits_{i=0}^{n} \dfrac{ {^i}f(x_0)}{i!} · (x - x_0)^{i} + R_{n}(x) = f(x_0) + \dfrac{ {^1}f(x_0)}{1!} · (x - x_0)^{1} + \dfrac{ {^2}f(x_0)}{2!} · (x - x_0)^2 + … + \dfrac{ {^n}f(x_0)}{n!} · (x - x_0)^n + \mathrm{R}_{n}(x)$

$f(x) = \sum\limits_{i=0}^{n} \dfrac{\mathrm{d}^{i} f(x_0)}{1! · \mathrm{d}^{i} x_0} · (x - x_0)^{i} + R_{n}(x) = f(x_0) + \dfrac{\mathrm{d}^{1} f(x_0)}{1! · \mathrm{d}^1 x_0} · (x - x_0)^{1} + \dfrac{\mathrm{d}^{2} f(x_0)}{2! · \mathrm{d}^2 x_0} · (x - x_0)^{2} + … + \dfrac{\mathrm{d}^{n} f(x_0)}{n! · \mathrm{d}^{n} x_0} · (x - x_0)^{n} + \mathrm{R}_{n}(x)$

$\mathrm{R}{n}(x) = o(x - x_0)^{n} = f(x) - f(x_0) - \sum\limits{i=1}^{n} \dfrac{ {^i}f(x_0)}{i!} · (x - x_0)^{i} \mathop{====}\limits^{∃θ∈(x_0, x)} \dfrac{ {^{n+1} } f(θ)}{(n + 1)!} · (x - x_0)^{n+1} \mathop{====}\limits^{∃θ∈(x_0, x)} \dfrac{ {^{n+1} } f(θ)}{n!} · (x - θ)^{n} · (x - x_0)^{1} \mathop{====}\limits^{∃θ∈(x_0, x)} \int_{x_0}^{x} \dfrac{ {^{n+1} } f(t)}{n!} · (x - t)^{n} \mathrm{d} t$

$⇓$ ${^m}\mathrm{R}n(x) \mathop{====}\limits{0≤m≤n} {^m}f(x) - {^m}f(x_0) - \sum\limits_{i=m+1}^{n} \dfrac{ {^i}f(x_0)}{i!} · (x - x_0)^{i-m} · \dfrac{i!}{(i-m)!} = {^m}f(x) - {^m}f(x_0) - \sum\limits_{i=m+1}^{n} \dfrac{ {^i}f(x_0)}{(i - m)!} · (x - x_0)^{i-m}$ $⇒$ $\lim\limits_{x⇝x_0} {^m}\mathrm{R}_n(x) ⇝ 0$
$⇓$ $\lim\limits_{x⇝x_0} \dfrac{\mathrm{R}{n}(x)}{(x - x_0)^{n} } = \lim\limits{x⇝x_0} \dfrac{(n - m)!}{n!} · \dfrac{ {^m}\mathrm{R}n(x)}{(x - x_0)^{n-m} } = \ldots = \lim\limits{x⇝x_0}\dfrac{1!}{n!} · \dfrac{^{n-1}\mathrm{R}n(x)}{(x - x_0)^1} = \lim\limits{x⇝x_0} \dfrac{0!}{n!} · \dfrac{ {^n}\mathrm{R}_n(x)}{(x - x_0)^{0} } ⇝ 0$ $⇒$ $\mathrm{R}_n(x) = o(x - x_0)^n$
$⇓$ $F (t) ≡ \sum\limits_{i=0}^n \dfrac{ {^i} f (t)}{i!} · (x - t)^i$ $⇒$ $\mathrm{R}_n (x) = F (x) - F (x_0)$    
$⇓$ ${^1} F (t) = \sum\limits_{i=0}^n \left[ \dfrac{ {^{i+1} } f (t)}{i!} · (x - t)^i - \dfrac{ {^i} f (t)}{i!} · i · (x - t)^{i-1} \right]$        
$⇓$ $^{1}F(t) = \sum\limits_{i=1}^{n+1} \dfrac{ {^i}f(t)}{(i - 1)!} · (x - t)^{i-1} - \sum\limits_{i=1}^{n} \dfrac{ {^i}f(t)}{(i-1)!} · (x - t)^{i-1} = \dfrac{ {^{n+1} }f(t)}{n!} · (x - t)^{n}$        
$⇓$ $\dfrac{R_n (x)}{G (x) - G (x_0)} = \dfrac{F (x) - F (x_0)}{G (x) - G (x_0)} = \left.\dfrac{ {^1} F (t)}{ {^1} G (t)}\right _{∃θ∈(x_0, x)} = \dfrac{ {^{n+1} } f (θ)}{n!} · \dfrac{(x - θ)^n}{ {^1} G (θ)} $      
$⇓$ $\mathrm{R}_n (x) \mathop{====}\limits^{∃θ∈(x_0, x)} \dfrac{ {^{n+1} } f (θ)}{n!} · \dfrac{(x - θ)^n}{ {^1} G (θ)} · [G (x) - G (x_0)]$        
$⇓$ $\mathrm{R}_n (x) \mathop{====}\limits^{∃θ∈(x_0, x)} \dfrac{ {^{n+1}f (θ)} }{(n + 1)!} · (x - x_0)^{n + 1}$ $⇐$ $G (t) ≡ (x - t)^{n + 1}$ ${^1}G (t) = -(n + 1) · (x - t)^n$  
$⇓$ $\mathrm{R}_n (x) \mathop{====}\limits^{∃θ∈(x_0, x)} \dfrac{ {^{n+1}f (θ)} }{n!} · (x - θ)^n · (x - x_0)^1$ $⇐$ $G (t) ≡ (x - t)^1$ ${^1}G (t) = -1$  
$⇓$ $\mathrm{R}n (x) \mathop{====}\limits^{∃θ∈(x_0, x)} \int\limits{x_0}^x \dfrac{ {^{n+1} }f (t)}{n!} · (x - t)^n \mathrm{d} t$ $⇐$ $G (t) ≡ \int\limits_{x_0}^t \dfrac{ {^{n+1} }f (t)}{n!} · (x - t)^n \mathrm{d} t$ ${^1}G (t) = \dfrac{ {^{n+1} }f (t)}{n!} · (x - t)^n$  

典例:若函数$f(x)$在区间$X$上二阶导函数存在。

$\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} = \lim\limits_{h⇝0} \dfrac{f(x+h) + f(x-h) - 2 · f(x)}{h^2}$

$⇓$ $f(x+h) = f(x) + \dfrac{\mathrm{d} f(x)}{1! · \mathrm{d} x} · h + \dfrac{\mathrm{d}^2 f(x)}{2! · \mathrm{d}^2 x} · h^2 + o_{+}(h^2)$ $f(x-h) = f(x) - \dfrac{\mathrm{d} f(x)}{1! · \mathrm{d} x} · h + \dfrac{\mathrm{d}^2 f(x)}{2! · \mathrm{d}^2 x} · h^2 + o_{-}(h^2)$
$⇓$ $\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} = \dfrac{f(x+h) + f(x-h) - 2 · f(x)}{h^2} + \left[ o_{+}(h^2) +o_{-}(h^2) \right]$  
$⇓$ $\dfrac{\mathrm{d}^2 f(x)}{\mathrm{d}^2 x} = \lim\limits_{h⇝0} \dfrac{f(x+h) + f(x-h) - 2 · f(x)}{h^2}$  

典例:函数$g(x)$与函数$f(x)$之比的极限值。

$\lim\limits_{x⇝x_0} \dfrac{g(x)}{f(x)} \mathop{======}\limits_{ {^h}f(x_0)\mathop{==}\limits^{h<n}0}^{ {^h}g(x_0)\mathop{==}\limits^{h<m}0} \lim\limits_{x⇝x_0} \dfrac{\dfrac{ {^m}g(x_0)}{m!}·(x - x_0)^{m} + o(x-x_0)^{m} }{\dfrac{ {^n}f(x_0)}{n!}·(x - x_0)^{n} + o(x-x_0)^{n} } = \lim\limits_{x⇝x_0} (x - x_0)^{m-n} · \dfrac{\dfrac{ {^m}g(x_0)}{m!} + \dfrac{o(x - x_0)^m}{(x - x_0)^m} }{\dfrac{ {^n}f(x_0)}{n!} + \dfrac{o(x - x_0)^n}{(x - x_0)^n} } ⇝ \mathop{0}\limits_{m>n};\mathop{\dfrac{ {^n}g(x_0)}{ {^n}f(x_0)} }\limits_{m=n};\mathop{∞}\limits_{m<n}$

典例:函数$f(x) = x^{n+1} · \mathcal{Q}(x)$在点$x_0=0$处连续但仅有一阶导数值,因此在点$x_0=0$处可唯一展开成$1$阶幂级数。

${^1}f(0) = \lim\limits_{x⇝0} \dfrac{f(x) - f(0)}{x - 0} = \lim\limits_{x⇝0} \dfrac{x^{n+1}·\mathcal{Q}(x)}{x} ⇝ 0$

$f(x) = f(0) + \dfrac{ {^1}f(0)}{1!} · (x - 0)^1 + R_{1}(x)= o(x^1)$

典例:函数$f(x) = \tan x$在点$x_0 = 0$处展开成$5$阶幂级数。

$\tan x = 1·x^1 + \dfrac{1}{3}·x^3 + \dfrac{2}{15}·x^5 + o(x^5)$

$f(x)$ $\dfrac{ {^0}f(0)}{0!}·x^0 + \dfrac{ {^1}f(0)}{1!}·x^1 + \dfrac{ {^2}f(0)}{2!}·x^2 + \dfrac{ {^3}f(0)}{3!}·x^3 + \dfrac{ {^4}f(0)}{4!}·x^4 + \dfrac{ {^5}f(0)}{5!}·x^5 + o(x^5)$
${^0}f(0)$ $\left[ \tan x \right]_{x=0} = 0$
${^1}f(0)$ $\left[ \cos^{-2} x \right]_{x=0} = 1$
${^2}f(0)$ $\left[ 2·\cos^{-3} x · \sin^{1} x \right]_{x=0} = 0$
${^3}f(0)$ $\left[ 6·\cos^{-4} x · \sin^{2} x + 2·\cos^{-2} x \right]_{x=0} = 2$
${^4}f(0)$ $\left[ 24·\cos^{-5} x · \sin^{3} x + 16·\cos^{-3} x · \sin^{1} x \right]_{x=0} = 0$
${^5}f(0)$ $\left[ 120·\cos^{-6} x · \sin^{4} x + 120·\cos^{-4} x · \sin^{2} x + 16·\cos^{-2} x \right]_{x=0} = 16$

典例:数列$s_n = \left(1 + \dfrac{0}{n^2}\right) · \left(1 + \dfrac{1}{n^2}\right) ··· \left(1 + \dfrac{n}{n^2}\right)$的极限值。

$\lim\limits_{n⇝∞^{+} } s_n = \lim\limits_{n⇝∞^{+} } \left(1 + \dfrac{0}{n^2}\right) · \left(1 + \dfrac{1}{n^2}\right) ··· \left(1 + \dfrac{n}{n^2}\right) ⇝ ә^{\frac{1}{2} }$

$⇓$ $x - \dfrac{x^2}{2} < \ln(1 + x) < x$ $⇐$ $\ln(1+x) = \sum\limits_{i=\rlap{≡}{0,}1}^{∞^{+} } \dfrac{(-1)^{i-1} }{i} · x^i$
$⇓$ $\sum\limits_{i=0}^{n} \dfrac{i}{n^2} - \sum\limits_{i=0}^{n} \dfrac{i^2}{2·n^{4} } < \sum\limits_{i=0}^{n} \ln\left(1 + \dfrac{i}{n^2}\right) < \sum\limits_{i=0}^{n} \dfrac{i}{n^2}$ $⇐$ $x = \dfrac{i}{n^2}$
$⇓$ $\dfrac{1}{n^2} · \left[ \dfrac{(n+1)^2}{2} - \dfrac{(n+1)^1}{2} \right] - \dfrac{1}{2·n^4} · \left[ \dfrac{(n+1)^3}{3} - \dfrac{(n+1)^2}{2} + \dfrac{(n+1)^1}{6} \right] < \ln s_n < \dfrac{1}{n^2} · \left[ \dfrac{(n+1)^2}{2} - \dfrac{(n+1)^1}{2} \right]$ $⇐$ $\sum\limits_{j=0}^{m} j^{1} = \dfrac{(m + 1)^{2} }{2} - \dfrac{(m + 1)^{1} }{2}$
$⇓$ $\lim\limits_{n⇝∞^{+} } s_n = \lim\limits_{n⇝∞^{+} } \left(1 + \dfrac{0}{n^2}\right) · \left(1 + \dfrac{1}{n^2}\right) ··· \left(1 + \dfrac{n}{n^2}\right) ⇝ ә^{\frac{1}{2} }$ $⇐$ $\sum\limits_{j=0}^{m} j^{2} = \dfrac{(m + 1)^3}{3} - \dfrac{(m + 1)^{2} }{2} + \dfrac{(m + 1)^{1} }{6}$

原函数

若存在函数$F(x)$,其导函数为$f(x)$,则称函数$F(x)$为函数$f(x)$的原函数。函数$f(x)$的全部原函数$F(x)$组成原函数族$\lbrace \tilde{F}(x) \rbrace = \left\lbrace ∀F; F(x) : \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = f(x) \right\rbrace$。

若函数$f(x)$存在原函数$F(x)$,则对函数$f(x)$作不定积分可得到函数$F(x)$的函数族$\lbrace \tilde{F}(x) \rbrace$。函数族$\lbrace \tilde{F}(x) \rbrace$中任意两个原函数$F_1(x)$与$F_2(x)$仅相差一个常数。

从几何上来看,函数$f(x)$的原函数族$\lbrace \tilde{F}(x) \rbrace$形成一系列沿纵坐标轴平移的曲线族,且曲线族在横坐标相同点处切线的斜率$\dfrac{\mathrm{d} }{\mathrm{d} x} [F(x) + \mathrm{Con}]$即为函数$f(x)$。

$\left[ f(x) = \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = \dfrac{\mathrm{d} }{\mathrm{d} x} [F(x) + \mathrm{Con}] \right] ⇔ \left[ \int f(x) \mathrm{d}x = \int \dfrac{\mathrm{d} F(x)}{\mathrm{d} x} \mathrm{d} x = F(x) + \mathrm{Con} \right]$

若函数$f(x)$的原函数$F(x)$满足特定的初始条件,则可唯一确定此原函数$F(x)$。

$\left[ y_0 = F(x_0) + \mathrm{Con} \right] ⇒ \left[ \mathrm{Con} = y_0 - F(x_0) \right]$

典例:质点作匀加速直线运动,加速度为$a = \dfrac{\mathrm{d} v(t)}{\mathrm{d} t} = \dfrac{\mathrm{d}^2 x(t)}{\mathrm{d}^2 t}$,初始速度为$v_0 = v(t_0)$,初始位移为$x_0 = x(t_0)$。

$v(t) = a·(t-t_0) + v_0$

$x(t) = \dfrac{a}{2} · (t - t_0)^2 + v_0 · (t - t_0) + x_0$

$v(t) = \int \dfrac{\mathrm{d} v(t)}{\mathrm{d} t} \mathrm{d} t = \int a \mathrm{d}t = a·t + \mathrm{Con} = a·(t-t_0) + v_0$ $⇐$ $v_0 = v(t_0) = a·t_0+\mathrm{Con}$
$x(t) = \int \dfrac{\mathrm{d} x(t)}{\mathrm{d}t} \mathrm{d}t = \int v(t) \mathrm{d}t = \int [a · (t - t_0) + v_0] \mathrm{d} x = \dfrac{a}{2}·(t-t_0)^2 + v_0·t + \mathrm{Con} = \dfrac{a}{2} · (t - t_0)^2 + v_0 · (t - t_0) + x_0$ $⇐$ $x_0 = x(t_0) = \dfrac{a}{2}·(t - t_0)^2 + v_0·t+\mathrm{Con}$

不定积分的运算性质。

$\int \mathrm{Con} · f(x) \mathrm{d}x = \mathrm{Con} · \int f(x) \mathrm{d} x$

$\int [f(x) + g(x)] \mathrm{d}x = \int f(x) \mathrm{d} x + \int g(x) \mathrm{d} x$

$\int [f(x) - g(x)] \mathrm{d}x = \int f(x) \mathrm{d} x - \int g(x) \mathrm{d} x$

     
$\dfrac{\mathrm{d} }{\mathrm{d} x} [\int \mathrm{Con} · f(x) \mathrm{d} x] = \mathrm{Con} · f(x)$ $=$ $\dfrac{\mathrm{d} }{\mathrm{d} x} [\mathrm{Con} · \int f(x) \mathrm{d} x] = \mathrm{Con} · \dfrac{\mathrm{d} }{\mathrm{d}x} [\int f(x) \mathrm{d} x] = \mathrm{Con} · f(x)$
$\dfrac{\mathrm{d} }{\mathrm{d} x} \int [f(x) + g(x)]\mathrm{d}x = f(x) + g(x)$ $=$ $\dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \int f(x) \mathrm{d} x + \int g(x) \mathrm{d} x \right] = \dfrac{\mathrm{d} }{\mathrm{d} x} \int f(x) \mathrm{d}x + \dfrac{\mathrm{d} }{\mathrm{d} x} \int g(x) \mathrm{d} x = f(x) + g(x)$
$\dfrac{\mathrm{d} }{\mathrm{d} x} \int [f(x) - g(x)]\mathrm{d}x = f(x) - g(x)$ $=$ $\dfrac{\mathrm{d} }{\mathrm{d} x} \left[ \int f(x) \mathrm{d} x - \int g(x) \mathrm{d} x \right] = \dfrac{\mathrm{d} }{\mathrm{d} x} \int f(x) \mathrm{d}x - \dfrac{\mathrm{d} }{\mathrm{d} x} \int g(x) \mathrm{d} x = f(x) - g(x)$

对于任意函数$f(x)$,未必都存在原函数$F(x)$,使得$\dfrac{\mathrm{d} F(x)}{\mathrm{d} x} = f(x)$。

典例:有理函数$\mathcal{Q} (x) = \mathop{1}\limits_{x∈ℚ}; \mathop{0}\limits_{x∉ℚ}$不存在原函数$F(x)$,使得$\dfrac{\mathrm{d} F(x)}{\mathrm{d}x} = \mathcal{Q}(x)$。

若函数$f(x)$在区间$X$上的导函数存在,则函数$f(x)$的导函数$\dfrac{\mathrm{d} f(x)}{\mathrm{d} x}$在区间$X$上不存在跳跃间断点。$P⇒Q$

若函数$f(x)$在区间$X$上含有跳跃间断点,则函数$f(x)$不存在原函数$F(x)$,使得$\dfrac{\mathrm{d}F(x)}{\mathrm{d}x} = f(x)$。$¬Q⇒¬P$

$⇓$ $\dfrac{\mathrm{d} f(x_0^{-})}{\mathrm{d} x_0^{-} } = \dfrac{\mathrm{d} f(x_0)}{\mathrm{d} x_0} = \dfrac{\mathrm{d} f(x_0^{+})}{\mathrm{d} x_0^{+} }$ 假设在点$x_0$处导数值存在    
$⇓$ $\dfrac{\mathrm{d} f(x_0^{-})}{\mathrm{d} x_0^{-} } = \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0^{-} } ≠ \left. \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} \right _{x_0^{+} } = \dfrac{\mathrm{d} f(x_0^{+})}{\mathrm{d} x_0^{+} }$ $\mathrm{False}$

定积分

将区间$[x_α,x_β]$划分成$n+1$个子区间$[x_i,x_{i+1}]$,每个子区间长度为$Δx_i = x_{i+1} - x_i$,任取子区间内点$θ_i∈[x_i,x_{i+1}]$,有函数$f(x)$在区间$[x_α,x_β]$上的定积分和。

$\sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i,x_{i+1}]} f(x) · Δx_i ≤ \sum\limits_{i=0}^{n} f(θ_i) · Δx_i ≤ \sum\limits_{i=0}^{n} \sup\limits_{x∈[x_{i},x_{i+1}]} f(x) · Δx_i$

对于函数$f(x)$,在将区间$[x_α,x_β]$划分成$n+1$个子区间$[x_i,x_{i+1}]$的基础上,多添加$m$个点再形成$m$个子区间,则其下积分和不减,且其上积分和不增。

对于函数$f(x)$,有任意多种将区间$[x_α,x_β]$划分成子区间的情形,但其下积分和始终小于等于其上积分和。此可理解为将任意两种划分情形互相叠加。

对于函数$f(x)$,无论在区间$[x_α,x_β]$上有任意多种划分情形,若其下积分和的极限等于上积分和的极限,则其积分和的极限存在且唯一,此即定积分。

$\sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i,x_{i+1}]} f(x) · Δx_i ≤ \sum\limits_{j=0}^{n+m} \inf\limits_{x∈[x_j,x_{j+1}]} f(x) · Δx_j ≤ \sum\limits_{j=0}^{n+m} f(θ_j) ·Δx_j ≤ \sum\limits_{j=0}^{n+m} \sup\limits_{x∈[x_{j},x_{j+1}]} f(x) · Δx_j ≤ \sum\limits_{i=0}^{n} \sup\limits_{x∈[x_i,x_{i+1}]} f(x) · Δx_i$

$\sum\limits_{i=0}^{n} \inf\limits_{x∈[x_{i},x_{i+1}]} f (x) · Δx_i ≤ \rlap{≡≡≡≡≡≡≡≡≡≡≡≡≡}{\sum\limits_{j=0}^{n+m} \inf\limits_{x∈[x_{j},x_{j+1}]} f (x) · Δx_j} ≤ \sum\limits_{j=0}^{n+m} f (θ_j) · Δx_j ≤ \rlap{≡≡≡≡≡≡≡≡≡≡≡≡≡}{\sum\limits_{j=0}^{n+m} \sup\limits_{x∈[x_{j},x_{j+1}]} f (x) · Δx_j} ≤ \sum\limits_{k=0}^{m} \sup\limits_{x∈[x_{k},x_{k+1}]} f (x) · Δx_k$

$\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_{i},x_{i+1}]} f (x) · Δx_i ≤ \rlap{≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡}\varlimsup\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_{i},x_{i+1}]} f (x) · Δx_i ≤ \lim\limits_{n+m⇝∞⁺}^{Δx_j⇝0} \sum\limits_{j=0}^{n+m} f (θ_j) · Δx_j ≤ \rlap{≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡≡}\varliminf\limits_{m⇝∞⁺}^{Δx_k⇝0} \sum\limits_{k=0}^{m} \sup\limits_{x∈[x_{k},x_{k+1}]} f (x) · Δx_k ≤ \lim\limits_{m⇝∞⁺}^{Δx_k⇝0} \sum\limits_{k=0}^{m} \sup\limits_{x∈[x_{k},x_{k+1}]} f (x) · Δx_k$

$⇓$ $[x_{i}, x_{i+1}] = [x_{j}, x_{j+1}] ∪ [x_{j+1}, x_{j+2}]$ $Δx_i = Δx_{j} + Δ_{j+1}$
$⇓$ $\inf\limits_{x∈[x_{i}, x_{i+1}]} f (x) ≤ \inf\limits_{x∈[x_{j}, x_{j+1}]} f (x), \inf\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) ≤ f (x_{j+1}) ≤ \sup\limits_{x∈[x_{j}, x_{j+1}]} f (x), \sup\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) ≤ \sup\limits_{x∈[x_i, x_{i+1}]} f (x)$  
$⇓$ $\inf\limits_{x∈[x_i, x_{i+1}]} f (x) · Δx_i ≤ \inf\limits_{x∈[x_j, x_{j+1}]} f (x) · Δx_{j} + \inf\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) · Δx_{j+1}$  
$⇓$ $\sup\limits_{x∈[x_j, x_{j+1}]} f (x) · Δx_{j} + \sup\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) · Δx_{j+1} ≤ \sup\limits_{x∈[x_i, x_{i+1}]} f (x) · Δx_{i}$  

对于函数$f (x)$,在将区间$[x_α,x_β]$划分成$n + 1$个子区间$[x_{i}, x_{i+1}]$的基础上,多添加$m$个点再形成$m$个子区间,则其振幅积和不增,且其振幅加和不减。

$0 ≤ \sum\limits_{j=0}^{n+m} \left[ \sup\limits_{x∈[x_j,x_{j+1}]} f(x) - \inf\limits_{x∈[x_j,x_{j+1}]} f(x) \right] · Δx_{j} ≤ \sum\limits_{i=0}^{n} \left[ \sup\limits_{x∈[x_i,x_{i+1}]} f(x) - \inf\limits_{x∈[x_i,x_{i+1}]} f(x) \right] · Δx_i = \sum\limits_{i=0}^{n} w_i · Δx_i$

$\sum\limits_{j=0}^{n+m} \left[ \sup\limits_{x∈[x_j,x_{j+1}]} f(x) - \inf\limits_{x∈[x_j,x_{j+1}]} f(x) \right] ≥ \sum\limits_{i=0}^{n} \left[ \sup\limits_{x∈[x_i,x_{i+1}]} f(x) - \inf\limits_{x∈[x_i,x_{i+1}]} f(x) \right] = \sum\limits_{i=0}^{n} w_i ≥ 0$

$⇓$ $[x_{i}, x_{i+1}] = [x_{j}, x_{j+1}] ∪ [x_{j+1}, x_{j+2}]$
$⇓$ $\inf\limits_{x∈[x_{i}, x_{i+1}]} f (x) ≤ \inf\limits_{x∈[x_{j}, x_{j+1}]} f (x), \inf\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) ≤ f (x_{j+1}) ≤ \sup\limits_{x∈[x_{j}, x_{j+1}]} f (x), \sup\limits_{x∈[x_{j+1}, x_{j+2}]} f (x) ≤ \sup\limits_{x∈[x_i, x_{i+1}]} f (x)$
$⇓$ $\left[ \sup\limits_{x∈[x_{j},x_{j+1}]} f (x) - \inf\limits_{x∈[x_{j},x_{j+1}]} f (x) \right] + \left[ \sup\limits_{x∈[x_{j+1},x_{j+2}]} f (x) - \inf\limits_{x∈[x_{j+1},x_{j+2}]} f (x) \right] = \left[ \sup\limits_{x∈[x_{j},x_{j+1}]} f (x) - \inf\limits_{x∈[x_{j+1},x_{j+2}]} f (x) \right] + \left[ \sup\limits_{x∈[x_{j+1},x_{j+2}]} f (x) - \inf\limits_{x∈[x_{j},x_{j+1}]} f (x) \right] ≥ 0 + 0$
$⇓$ $\left[ \sup\limits_{x∈[x_{j},x_{j+1}]} f (x) - \inf\limits_{x∈[x_{j},x_{j+1}]} f (x) \right] + \left[ \sup\limits_{x∈[x_{j+1},x_{j+2}]} f (x) - \inf\limits_{x∈[x_{j+1},x_{j+2}]} f (x) \right] ≥ \left[ \sup\limits_{x∈[x_{i},x_{i+1}]} f (x) - \inf\limits_{x∈[x_{i},x_{i+1}]} f (x) \right]$

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则其下积分和的极限等于上积分和的极限,反之亦然。

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则其振幅积和的极限为零,反之亦然。

$\int\limits_{x_α}^{x_β} f(x) · \mathrm{d}x \mathop{≡≡≡≡}\limits_{x_{α}=x_0}^{x_{β}=x_{n+1} } \lim\limits_{n⇝∞^{+} }^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i,x_{i+1}]} f(x) · Δx_i = \lim\limits_{n⇝∞^+}^{Δx_i⇝0} \sum\limits_{i=0}^{n} f(θ_i) · Δx_i = \lim\limits_{n⇝∞^{+} }^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{x∈[x_i,x_{i+1}]} f(x) · Δx_i$

$0 ⇜ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n w_i · Δx_i ≡ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n \sup\limits_{u,v∈[x_{i}, x_{i+1}]} f (u) - f (v) · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \left[ \sup\limits_{x∈[x_{i},x_{i+1}]} f (x) - \inf\limits_{x∈[x_i,x_{i+1}]} f (x) \right] · Δx_i$

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则可任取子区间内点$θ_i$处函数值$f(θ_i)$为该子区间的平均值。

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则可将区间$[x_α,x_β]$按等间距$\dfrac{x_β-x_α}{n + 1}$划分成$n+1$个子区间。

若函数$f (x)$在区间$[x_α,x_β]$上的定积分存在,则当子区间的间距趋于零时,定积分与子区间的取值无关,定积分与子区间的划分无关。

$\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n f (θ_i) · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n f (x_i) · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n f (x_{i+1}) · Δx_i = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n f \left[ i · \dfrac{x_β - x_α}{n + 1} \right] · \dfrac{x_β - x_α}{n + 1} = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} f \left[ (i + 1) · \dfrac{x_β - x_α}{n + 1} \right] · \dfrac{x_β - x_α}{n + 1}$

$\left[ 0 ⇜ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n w_i · Δx_i \right] ⇒ \left[ 0 ⇜ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n w_i · \dfrac{x_β - x_α}{n + 1} \right] ⇒\left[ 0 ⇜ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^n \sup\limits_{u,v∈[x_i, x_{i+1}]} f (u) - f (v) · \dfrac{x_β - x_α}{n + 1} \right] ⇒ \left[ 0 ⇜ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} f (x_{i+1}) - f (x_{i}) · \dfrac{x_β - x_α}{n + 1} \right]$

若函数$f(X)$在区间$[x_α,x_β]$上连续因此一致连续,则其振幅积和的极限为零。连续函数$f(x)$在区间$[x_α,x_β]$上的定积分必存在。

$⇓$ $∀ε>0;∃δ>0;∀u,v∈[x_α,x_β]; u-v ≤δ ⇒ f(u) - f(v) ≤ ε$ $⇔$ $\lim\limits_{x⇝x_t} f (x) \mathop{↭}\limits_{x,x_t∈[x_α,x_β]} f (x_t)$
$⇓$ $∀ε>0;∃δ=\max \lbraceΔx_i\rbrace;\sup\limits_{u,v∈[x_i,x_{i+1}]} f(u)-f(v) ≤ ε$        
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) - f (v) · Δx_i < ε · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} Δx_i = ε · (x_β-x_α) ⇝ 0$        

若函数$f (x)$在区间$[x_α,x_β]$上的定积分存在,则函数$f (x)$在区间$[x_α,x_β]$上有确界,反之不对。

$\left[ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i ⇝ 0 \right] ⇒ \left[ f (x) ≤ \mathrm{Sup} \right]$

$⇓$ $0 ⇜ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i$ $⇒$ $0 ⇜ \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} w_i · \dfrac{x_β - x_α}{n + 1}$            
$⇓$ $0 ⇜ \lim\limits_{n⇝∞⁺} \dfrac{\sup\limits_{x∈[x_α,x_β]} f (x) - \inf\limits_{x∈[x_α,x_β]} f (x)}{n + 1} · (x_β - x_α)$ $⇐$ $\sum\limits_{i=0}^n w_i = \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} f (u) - f (v) ≥ \sup\limits_{x∈[x_α,x_β]} f (x) - \inf\limits_{x∈[x_α,x_β]} f (x) ≥ 0$        
$⇓$ $\sup\limits_{x∈[x_α,x_β]} f (x) - \inf\limits_{x∈[x_α,x_β]} f (x) ≠ ∞⁺$                
$⇓$ $\left[ \sup\limits_{x∈[x_α,x_β]} f (x) ≠ ∞⁺ \right] ∧ \left[ \inf\limits_{x∈[x_α,x_β]} f (x) ≠ ∞⁻ \right]$ $⇒$ $ f(x) ≤ \mathrm{Sup} = \max\lbrace \inf f(x) , \sup f(x) \rbrace$

反例:函数$\mathcal{Q} (x) = \mathop{1}\limits_{x∈ℚ}; \mathop{0}\limits_{x∉ℚ}$在区间$[0, 1]$上有确界,但其定积分不存在。

$\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i, x_i+1]} \mathcal{Q} (x) · Δx_i = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} 0 · Δx_i ⇝ 0 ≠ 1 ⇜ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} 1 · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{x∈[x_i, x_{i+1}]} \mathcal{Q} (x) · Δx_i$

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则函数$f(x)$在区间$[x_α,x_β]$上的断点集为零测集,反之亦然。函数$f (x)$在区间$[x_α, x_β]$上几乎处处连续。

$\left[ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i ⇝ 0 \right] ⇔ \left[ ∀ε>0,𝜁>0; \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i≥ε} < 𝜁 \right]$

$⇓$ $∀ε>0,𝜁>0;$ $ε · 𝜁 > \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n w_i · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{w_i · Δx_i}\limits_{w_i<ε} + \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n \mathop{w_i · Δx_i}\limits_{w_i≥ε} ≥ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{w_i · Δx_i}\limits_{w_i≥ε} ≥ ε · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i≥ε}$
$⇓$ $∀ε>0,𝜁>0;$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i≥ε} < 𝜁$
$⇓$ $∀ε>0,𝜁>0;$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{w_i · Δx_i}\limits_{w_i<ε} + \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{w_i · Δx_i}\limits_{w_i≥ε} ≤ ε · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i<ε} + \sup\limits w_i · \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i≥ε} ≤ ε · (x_β - x_α) + \sup\limits w_i · 𝜁$
$⇓$ $∀ε>0,𝜁>0;$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i ⇝ 0$
特例:函数$\mathcal{R} (x) \mathop{======}\limits_{m∈ℤ;n∈ℤ^+}^{1 = \gcd ( m , n )} \mathop{\dfrac{1}{n} }\limits_{x=\frac{m}{n} }; \mathop{0}\limits_{x≠\frac{m}{n} }$在区间$[0, 1]$上,任何无理点处连续,任何有理点处不连续,其断点集为零测集,因此其定积分存在。

$\int\limits_{0}^{1} \mathcal{R} (x) \mathrm{d} x = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i,x_{i+1}]} \mathcal{R} (x) · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} 0 · Δx_i ⇝ 0$

若函数$f^{⤨} (x)$在区间$[x_α, x_β]$上单调有确界,则其断点集为零测集,因此其定积分存在。

$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i ≤ ε · \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} w_i = ε · \left[ \sup\limits_{x∈[x_α,x_β]} f^{⤨} (x) - \inf\limits_{x∈[x_α,x_β]} f^{⤨} (x) \right]$
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i · Δx_i ⇝ 0$
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i≥ε} ⇝ 0$
若函数$f (x)$在区间$[x_α, x_β]$上的定积分存在,则函数$ f (x) $在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集,反之不对。

若函数$f (x)$以及函数$g (x)$在区间$[x_α, x_β]$上的定积分存在,则函数$f (x) + g (x)$在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集,反之不对。

若函数$f (x)$以及函数$g (x)$在区间$[x_α, x_β]$上的定积分存在,则函数$f (x) - g (x)$在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集,反之不对。

若函数$f (x)$以及函数$g (x)$在区间$[x_α, x_β]$上的定积分存在,则函数$f (x) · g (x)$在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集,反之不对。

若函数$f (x)$以及函数$g (x) ≠ 0$在区间$[x_α, x_β]$上的定积分存在,则函数$\dfrac{f (x)}{g (x)}$在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集,反之不对。

$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^n \sup\limits_{u,v∈[x_i,x_{i+1}]}   f (u) - f (v)   · Δx_i ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) - f (v) · Δx_i ⇝ 0$              
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_{i}^{ f } · Δx_i ⇝ 0$                        
$⇓$ $∃ε>0; \left[ w_i^f < ε \right] ⇒ \left[ w_i^{ f } ≤ w_i^f < ε \right]$                        
$⇓$ $∃ε>0; \left[ ε ≤ w_i^{ f } \right] ⇒ \left[ ε ≤ w_i^{f} \right]$                        
$⇓$ $\mathcal{O}^{ f } ⊆ \mathcal{O}^{f}$                        
                               
$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} [ f (u) + g (u) ] - [ f (v) + g (v) ] · Δx_i ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} f (u) - f (v) · Δx_i + \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (u) - g (v) · Δx_i ⇝ 0$                
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i^{f+g} ·Δx_i ⇝ 0$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ w_i^f < ε^f \right] ∧ \left[ w_i^g < ε^g \right] ⇒ \left[ w_i^{f+g} ≤ (w_i^f + w_i^g) < ε^f + ε^g = ε \right]$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ ε^f + ε^g = ε ≤ w_i^{f+g} \right] ⇒ \left[ ε^f ≤ w_i^{f} \right] ∨ \left[ ε^g ≤ w_i^g \right]$                            
$⇓$ $\mathcal{O}^{f+g} ⊆ \mathcal{O}^{f} ∪ \mathcal{O}^g$                            
                               
$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} [ f (u) - g (u) ] - [ f (v) - g (v) ] · Δx_i ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} f (u) - f (v) · Δx_i + \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (u) - g (v) · Δx_i ⇝ 0$                
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i^{f-g} ·Δx_i ⇝ 0$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ w_i^f < ε^f \right] ∧ \left[ w_i^g < ε^g \right] ⇒ \left[ w_i^{f-g} ≤ (w_i^f + w_i^g) < ε^f + ε^g = ε \right]$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ ε^f + ε^g = ε ≤ w_i^{f-g} \right] ⇒ \left[ ε^f ≤ w_i^{f} \right] ∨ \left[ ε^g ≤ w_i^g \right]$                            
$⇓$ $\mathcal{O}^{f-g} ⊆ \mathcal{O}^{f} ∪ \mathcal{O}^g$                            
                               
$⇓$ $\sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) · g (u) - f (v) · g (v) ≤ \sup\limits_{u,v∈[x_i,x_{i+1}]} [ f (u) · g (u) - g (v) + g (v) · f (u) - f (v) ] ≤ \sup f (x) · \sup\limits_{u,v∈[x_i,x_{i+1}]} g (u) - g (v) + \sup g (x) · \sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) - f (v) $
$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) · g (u) - f (v) · g (v) · Δx_i ≤ \sup f(x) · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_i,x_{i+1}]} g (u) - g (v) · Δx_i + \sup g (x) · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_i,x_{i+1}]} f (u) - f (v) · Δx_i ⇝ 0$        
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_{i}^{f·g} · Δx_i ⇝ 0$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ w_i^f < ε^f \right] ∧ \left[ w_i^g < ε^g \right] ⇒ \left[ w_i^{f·g} ≤ \left[ \sup f (x) · w_i^g + \sup g (x) · w_i^f \right] < \sup f (x) · ε^f + \sup g (x) · ε^g = ε \right]$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ ε ≤ w_i^{f·g} \right] ⇒ \left[ ε^f ≤ w_i^f \right] ∨ \left[ ε^g ≤ w_i^g \right]$                            
$⇓$ $\mathcal{O}^{f·g} ⊆ \mathcal{O}^{f} ∪ \mathcal{O}^{g}$                            
                               
$⇓$ $\sup\limits_{u,v∈[x_{i},x_{i+1}]} \left \dfrac{f (u)}{g (u)} - \dfrac{f (v)}{g (v)} \right = \sup\limits_{u,v∈[x_{i},x_{i+1}]} \left \dfrac{g (v) · [ f (u) - f (v) ] - f (v) · [ g (u) - g (v) ]}{g (u) · g (v)} \right ≤ \dfrac{1}{\inf \left g (x) \right } · \sup\limits_{u,v∈[x_{i},x_{i+1}]} f (u) - f (v) + \dfrac{\sup f (x) }{\inf g^2 (x) } · \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (u) - g (v) $
$⇓$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} \left \dfrac{f (u)}{g (u)} - \dfrac{f (v)}{g (v)} \right · Δx_i ≤ \dfrac{1}{\inf g (x) } · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} f (u) - f (v) · Δx_i + \dfrac{\sup f (x) }{\inf g^2 (x) } · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (u) - g (v) · Δx_i ⇝ 0$    
$⇓$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_i^{\frac{f}{g} } · Δx_i ⇝ 0$                            
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ w_i^f < ε^f \right] ∧ \left[ w_i^g < ε^g \right] ⇒ \left[ w_i^{\frac{f}{g} } ≤ \left[ \dfrac{1}{\inf g (x) } · w_i^f + \dfrac{\sup f (x) }{\inf g^2 (x) } · w_i^g \right] < \dfrac{1}{\inf g (x) } · ε^f + \dfrac{\sup f (x) }{\inf g^2 (x) } · ε^g = ε \right]$    
$⇓$ $∃ε>0;∃ε^f,ε^g>0; \left[ ε ≤ w_i^{\frac{f}{g} } \right] ⇒ \left[ ε^f ≤ w_i^f \right] ∨ \left[ ε^g ≤ w_i^g \right]$                            
$⇓$ $\mathcal{O}^{\frac{f}{g} } ⊆ \mathcal{\mathcal{O}^{f} } ∪ \mathcal{O}^{g}$                            
特例:函数$f (x) = \mathop{1}\limits_{x∈ℚ};\mathop{-1}\limits_{x∉ℚ}$在区间$[0, 1]$上的定积分不存在,其断点集非为零测集。但函数$ f (x) = 1$在区间$[0, 1]$上的定积分存在。

若函数$g(y)$在区间$[y_α, y_β]$上连续,且函数$y = f(x)$在区间$[x_α, x_β]$上的定积分存在,则复合函数$g(f(x))$在区间$[x_α, x_β]$上的定积分存在,其断点集为零测集。

若函数$g(y)$在区间$[y_α, y_β]$上的定积分存在,且函数$y = f(x)$在区间$[x_α, x_β]$上连续因此定积分存在,则复合函数$g(f(x))$在区间$[x_α, x_β]$上的定积分未必存在。

若函数$g(y)$在区间$[ y_α, y_β ]$上的定积分存在,且函数$y = f(x)$在区间$[x_α, x_β]$上的定积分存在,则复合函数$g(f(x))$在区间$[x_α, x_β]$上的定积分未必存在。

$⇓$   $\left[ \lim\limits_{y⇝y_0} g(y) \mathop{⇝}\limits_{y,y_0∈[x_α,x_β]} g(y_0) \right] ⇔ \left[ \lim\limits_{y↭y_t} g(y)\mathop{↭}\limits_{y,y_0∈[x_α,x_β]}g(y_t) \right]$                
$⇓$ $∀ε>0;∃δ>0;$ $\sup\limits_{ y-y_t ≤δ} g(y) - g(y_t) ≤ ε $        
$⇓$ $∀ε>0;∃δ>0;$ $\sup\limits_{ f(x)-f(x_t) ≤δ} g(f(x)) - g(f(x_t)) ≤ ε $        
$⇓$ $∀𝜁>0;∀δ>0;$ $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i^{f}>δ} < 𝜁$                
$⇓$ $∀ε,𝜁>0;∃δ>0;$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (f (u)) - g (f (v)) · Δx_i = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} \mathop{ g (f (u)) - g (f (v)) · Δx_i}\limits_{ f(u)-f(v) ≤w_i^{f}≤δ} + \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} \mathop{ g(f(u)) - g(f(v)) · Δx_i}\limits_{w_i^f>δ}$
$⇓$ $∀ε,𝜁>0;∃δ>0;$ $0 ≤ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{u,v∈[x_{i},x_{i+1}]} g (f(u)) - g(f(v)) · Δx_i < ε · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i^{f}≤δ} + \mathrm{Sup} · \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \mathop{Δx_i}\limits_{w_i^{f}>δ} < ε · (x_β - x_α) + \mathrm{Sup} · 𝜁 ⇝ 0$            
$⇓$   $\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} w_{i}^{g∘f} · Δx_i ⇝ 0$                
                     
$⇓$ $∀ε>0;∃δ>0;$ $\sup\limits_{ f(x)-f(x_t) ≤δ} g (f (x)) - g (f (x_t)) ≤ ε $        
$⇓$ $∃ε>0;∃δ>0;$ $[ w_i^{f} ≤ δ ] ⇒ [ w_i^{g∘f} ≤ ε ]$                
$⇓$ $∃ε>0;∃δ>0;$ $[ w_i^{g∘f} > ε ] ⇒ [ w_i^{f} > δ ]$                
$⇓$   $\mathcal{O}^{g∘f} ⊆ \mathcal{O}^{f}$                
反例:函数$\mathrm{sgn} (y) = \mathop{-1}\limits_{y<0};\mathop{0}\limits_{y=0};\mathop{+1}\limits_{y>0}$在区间$[0, 1]$上的定积分存在,函数$\mathcal{F} (x) ≡ \mathop{I_i - \left x - X_{i,j} \right }\limits_{ x - X_{i,j} < I_{i} }; \mathop{0}\limits_{ x - X_{i,j} ≥ I_{i} }$在区间$[0, 1]$上连续因此定积分存在。
但复合函数$\mathrm{sgn} (\mathcal{F} (x)) = \mathop{1}\limits_{ x-X_{i,j} <I_i};\mathop{0}\limits_{ x-X_{i,j} ≥I_i}$在区间$[0, 1]$上的断点集非为零测集,其定积分不存在。
反例:函数$\mathrm{sgn} (y) = \mathop{-1}\limits_{y<0};\mathop{0}\limits_{y=0};\mathop{+1}\limits_{y>0}$在区间$[0, 1]$上的定积分存在,函数$\mathcal{R} (x) \mathop{======}\limits_{m∈ℤ;n∈ℤ^+}^{1 = \gcd ( m , n )} \mathop{\dfrac{1}{n} }\limits_{x=\frac{m}{n} }; \mathop{0}\limits_{x≠\frac{m}{n} }$在区间$[0, 1]$上的定积分存在。

但复合函数$\mathrm{sgn} (\mathcal{R} (x)) = \mathcal{Q} (x) = \mathop{1}\limits_{x∈Q}; \mathop{0}\limits_{x∉Q}$在区间$[0, 1]$上的断点集非为零测集,其定积分不存在。

定积分的运算性质

定积分的积分区间下限与积分区间上限对换时,约定定积分的正负性对换。

$\int\limits_{x_α}^{x_β} f (x) \mathrm{d} x = - \int\limits_{x_β}^{x_α} f (x) \mathrm{d} x$                    
$\int\limits_{x_α}^{x_α} f (x) \mathrm{d} x = 0$ $⇐$ $\int\limits_{x_α}^{x_α} f (x) \mathrm{d} x = - \int\limits_{x_α}^{x_α} f (x) \mathrm{d} x$                
$\int\limits_{x_α}^{x_β} f (x) \mathrm{d} x = \int\limits_{x_α}^{x_γ} f (x) \mathrm{d} x + \int\limits_{x_γ}^{x_β} f (x) \mathrm{d} x$                    
$\int\limits_{x_α}^{x_β} \mathrm{Con} · f (x) \mathrm{d} x = \mathrm{Con} · \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x$                    
$\int\limits_{x_α}^{x_β} [ f (x) + g (x) ] \mathrm{d} x = \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x + \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x$                    
$\int\limits_{x_α}^{x_β} [ f (x) - g (x) ] \mathrm{d} x = \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x - \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x$                    
$[ f (x) ≤ g (x) ] ⇒ \left[ \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x ≤ \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x \right]$                    
$[ 0 ≤ g (x) ] ⇒ \left[ 0 ≤ \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x \right]$ $⇐$ $f (x) = 0$                
$[ f (x) ≤ g (x) ] ∧ \left[ \lim\limits_{x⇝x_0} f (x) ⇝ f (x_0) < g (x_0) ⇜ \lim\limits_{x⇝x_0} g (x) \right] ⇒ \left[ \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x < \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x \right]$ $⇐$ $\left[ \int\limits_{x_α}^{x_0-δ} + \int\limits_{x_0-δ}^{x_0+δ} + \int\limits_{x_0+δ}^{x_β} \right] [ f (x) - g (x) ] \mathrm{d} x < 0$                
$[ 0 ≤ g (x) ] ∧ \left[ 0 < g (x_0) ⇜ \lim\limits_{x⇝x_0} g (x) \right] ⇒ \left[ 0 < \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x \right]$ $⇐$ $f (x) = 0$                
$[ 0 ≤ g (x) ] ∧ \left[ 0 = \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x \right] ⇒ \left[ 0 = g (x_0) ⇜ \lim\limits_{x⇝x_0} g (x) \right]$ $⇐$ $[ P ∧ Q ⇒ R ] ⇔ [ P ∧ ¬R ⇒ ¬Q ]$                
$\left \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x \right ≤ \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x$ $⇐$ $- \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x ≤ \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x ≤ + \int\limits_{x_α}^{x_β} f (x) \mathrm{d} x$

微积分基本定理

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,则其变上限积分与其变下限积分存在,其变上限积分与其变下限积分可互相转换。

$\mathrm{U}{x_α}^{f}(x) = \int\limits{x_α}^{x} f(x) · \mathrm{d} x$

$\mathrm{L}{x_β}^{f}(x) = \int\limits{x}^{x_β} f(x) · \mathrm{d} x$

$\mathrm{U}{x_γ}^{f}(x) = \int\limits{x_γ}^{x} f(x)·\mathrm{d}x = -\int\limits_{x}^{x_γ} f(x)·\mathrm{d}x = -\mathrm{L}_{x_γ}^{f}(x)$

若函数$f(x)$在区间$[x_α,x_β]$上的定积分存在,因此函数$f(x)$在区间$[x_α,x_β]$上有确界,则其变上限积分$\mathrm{U}(x)$在区间$[x_α,x_β]$上连续。

$\left[ \int\limits_{x_α}^{x_β} f(x) · \mathrm{d} x = \lim\limits_{n⇝∞^+}^{Δx_i⇝0} \sum\limits_{i=0}^{n} f(θ_i) · Δx_i \right] ⇒ \left[ \lim\limits_{x⇝x_0} \mathrm{U}(x) ⇝ \mathrm{U}(x_0) \right]$

$⇓$ $\left[ \int\limits_{x_α}^{x_β} f(x) · \mathrm{d} x = \lim\limits_{n⇝∞^+}^{Δx_i⇝0} \sum\limits_{i=0}^{n} f(θ_i) · Δx_i \right] ⇒ \left[ f(x) ≤ \mathrm{Sup} \right]$                
$⇓$ $∀x_0∈[x_α,x_β];∀ε>0;∃δ>0;∀x∈[x_α,x_β]; \mathrm{U}(x) - \mathrm{U}(x_0) = \left \int\limits_{x_α}^{x} f(x) · \mathrm{d}x - \int\limits_{x_α}^{x_0} f(x) · \mathrm{d} x \right = \left \int\limits_{x_0}^{x} f(x) · \mathrm{d}x \right ≤ \mathrm{Sup} · \left \int\limits_{x_0}^{x} \mathrm{d}x \right = \mathrm{Sup}· (x - x_α) = \mathrm{Sup} · δ = ε$
$⇓$ $\lim\limits_{x⇝x_0} \mathrm{U}(x) ⇝ \mathrm{U}(x_0)$                

若函数$f(x)$在区间$[x_α,x_β]$上连续,则其变上限积分$\mathrm{U}(x)$为函数$f(x)$的原函数,且同理其变下限积分$\mathrm{L}(x)$为函数$f(x)$的原函数。

$\dfrac{\mathrm{d} }{\mathrm{d} x} \int\limits_{x_α}^{x} f(x) · \mathrm{d} x = \dfrac{\mathrm{d} \mathrm{U}(x)}{\mathrm{d}x} = f(x)$

$\dfrac{\mathrm{d} }{\mathrm{d}x} \int\limits_{v(x)}^{u(x)} f(x) ·\mathrm{d}x = \dfrac{\mathrm{d} \mathrm{U}(u(x))}{\mathrm{d} x} - \dfrac{\mathrm{d}\mathrm{U}(v(x))}{\mathrm{d}x} = f(u(x)) · {^1}u(x) - f(v(x)) · {^1} v(x)$

$⇓$ $∀x_0∈[x_α,x_β];∀ε>0;∃δ>0;∀x_t∈[x_α,x_β]; x_t - x_0 ≤ δ ⇒ f(x_t) - f(x_0) ≤ε$                    
$⇓$ $\left \dfrac{\mathrm{U}(x_t) - \mathrm{U}(x_0)}{x_t - x_0} - f(x_0) \right = \left \dfrac{\int\limits_{x_α}^{x_t} f(x) · \mathrm{d}x - \int\limits_{x_α}^{x_0} f(x)·\mathrm{d}x}{x_t - x_0} - f(x_0) \right = \left \dfrac{\int\limits_{x_0}^{x_t} f(x)·\mathrm{d}x - \int\limits_{x_0}^{x_t} f(x_0)·\mathrm{d}x}{x_t - x_0} \right ≤ \left \dfrac{\int\limits_{x_0}^{x_t} f(x)-f(x_0) ·\mathrm{d}x}{x_t - x_0} \right ≤ \dfrac{ε·\left \int\limits_{x_0}^{x_t} \mathrm{d}x \right }{ x_t - x_0 } = ε$
$⇓$ $\dfrac{\mathrm{d} }{\mathrm{d} x} \int\limits_{x_α}^{x} f(x) · \mathrm{d} x = \dfrac{\mathrm{d} \mathrm{U}(x)}{\mathrm{d}x} = f(x)$                            
                               
$⇓$ $\int\limits_{v(x)}^{u(x)} f(x) ·\mathrm{d}x = \int\limits_{x_γ}^{u(x)} f(x) · \mathrm{d}x + \int\limits_{v(x)}^{x_γ} f(x) · \mathrm{d}x = \int\limits_{x_γ}^{u(x)} f(x) · \mathrm{d} x - \int\limits_{x_γ}^{v(x)} f(x) · \mathrm{d} x = \mathrm{U}(u(x)) - \mathrm{U}(v(x))$                            
$⇓$ $\dfrac{\mathrm{d} }{\mathrm{d}x} \int\limits_{v(x)}^{u(x)} f(x) ·\mathrm{d}x = \dfrac{\mathrm{d} \mathrm{U}(u(x))}{\mathrm{d} x} - \dfrac{\mathrm{d}\mathrm{U}(v(x))}{\mathrm{d}x} = \dfrac{\mathrm{d}\mathrm{U}(u)}{\mathrm{d}u} · \dfrac{\mathrm{d}u(x)}{\mathrm{d}x} - \dfrac{\mathrm{d}\mathrm{U}(v)}{\mathrm{d}v} ·\dfrac{\mathrm{d}v(x)}{\mathrm{d}x} = f(u(x)) · {^1}u(x) - f(v(x)) · {^1} v(x)$                            

若函数$f(x)$在区间$[x_α,x_β]$上连续,则其在区间$[x_α,x_β]$上的定积分为其任意原函数在该两端点处的差值。【微积分基本定理】

$\int\limits_{x_α}^{x_β} f(x) · \mathrm{d}x = F(x_β) - F(x_α) = \left. F(x) \right _{x_α}^{x_β}$
$⇓$ $\int\limits_{x_α}^{x_β} f(x) · \mathrm{d}x \mathop{====}\limits_{\mathrm{U}(x_α) = 0} \mathrm{U}(x_β) - \mathrm{U}(x_α) = [\mathrm{U}(x_β) + \mathrm{Con}] - [\mathrm{U}(x_α)+\mathrm{Con}] = F(x_β) - F(x_α) = \left. F(x) \right _{x_α}^{x_β}$
     

典例:计算定积分$\int\limits_{0}^{π} \sin x · \mathrm{d}x$。

$\left[ \int\limits_{x_α}^{x_β} \sin x · \mathrm{d} x = [-\cos x]{x_α}^{x_β} \right] ⇒ \left[ \int\limits{0}^{π} \sin x · \mathrm{d}x = [-\cos x]_{0}^{π} = 2 \right]$

典例:计算极限值$\lim\limits_{n⇝∞^{+} } \dfrac{1^{p}+2^{p}+···+n^{p} }{n^{p+1} }$。

$\lim\limits_{n⇝∞^{+} } \dfrac{1^{p}+2^{p}+···+n^{p} }{n^{p+1} } = \lim\limits_{n⇝∞^{+} } \left[ \left(\dfrac{1}{n}\right)^{p} + \left( \dfrac{2}{n} \right)^{p} + ··· + \left( \dfrac{n}{n} \right)^{p} \right] · \dfrac{1}{n} \mathop{===}\limits^{p>-1} \int\limits_{0}^{1} x^{p} · \mathrm{d} x = \left[ \dfrac{x^{p+1} }{p+1} \right]_{0}^{1} = \dfrac{1}{p + 1}$

换元积分法。

$\int\limits_{x_α}^{x_β} g(f(x))·{^1}f(x) · \mathrm{d}x = \int\limits_{x_α}^{x_β} g(f(x)) · \mathrm{d} f(x) \mathop{====}\limits_{t_α=f(x_α)}^{t_β = f(x_β)} \int\limits_{t_α}^{t_β} g(t) · \mathrm{d}t$

分部积分法。

$\int\limits_{x_α}^{x_β} u(x) · \mathrm{d}v(x) = [u(x)·v(x)]{x_α}^{x_β} - \int\limits{x_α}^{x_β} v(x) ·\mathrm{d} u(x)$

$\int\limits_{x_α}^{x_β} u(x) ·{^1}v(x) · \mathrm{d}x = [u(x)·v(x)]{x_α}^{x_β} - \int\limits{x_α}^{x_β} {^1}u(x) · v(x) · \mathrm{d} x$

$⇓$ $\int\limits_{x_α}^{x_β} g(f(x)) · {^1}f(x) · \mathrm{d} x = \int\limits_{x_α}^{x_β} g(f(x)) · \dfrac{\mathrm{d} f(x)}{\mathrm{d} x} · \mathrm{d}x = \int\limits_{x_α}^{x_β} g(f(x)) · \mathrm{d} f(x) \mathop{====}\limits_{t_α=f(x_α)}^{t_β=f(x_β)} \int\limits_{t_α}^{t_β} g(t) · \mathrm{d} t$    
       
$⇓$ $\mathrm{d}[u(x) · v(x)] = u(x) · \mathrm{d} v(x) + v(x) · \mathrm{d} u(x)$ $⇔$ $\dfrac{\mathrm{d} }{\mathrm{d} x} [u(x) · v(x)] = \dfrac{\mathrm{d} u(x)}{\mathrm{d} x} · v(x) + u(x) · \dfrac{\mathrm{d} v(x)}{\mathrm{d} x}$
$⇓$ $[u(x) · v(x)]{x_α}^{x_β} = \int\limits{x_α}^{x_β} \mathrm{d}[u(x) · v(x)] = \int\limits_{x_α}^{x_β} u(x) · \mathrm{d} v(x) + \int\limits_{x_α}^{x_β} v(x) · \mathrm{d} u(x)$    
$⇓$ $\int\limits_{x_α}^{x_β} u(x) · \mathrm{d}v(x) = [u(x)·v(x)]{x_α}^{x_β} - \int\limits{x_α}^{x_β} v(x) ·\mathrm{d} u(x)$    
$⇓$ $\int\limits_{x_α}^{x_β} u(x) ·{^1}v(x) · \mathrm{d}x = [u(x)·v(x)]{x_α}^{x_β} - \int\limits{x_α}^{x_β} {^1}u(x) · v(x) · \mathrm{d} x$    

分部积分法的升降幂型式。

$\int\limits_{x_α}^{x_β} {^0}u(x) · {^{n+1} }v(x) · \mathrm{d}x = \left[ \sum\limits_{i=0}^{n} (-1)^{n} ·{^i}u(x) ·{^{n-i} }v(x) \right]{x_α}^{x_β} + (-1)^{n+1} \int\limits{x_α}^{x_β} {^{n+1} }u(x) · {^0}v(x) ·\mathrm{d}x$

$\int\limits_{x_α}^{x_β} {^0}u(x) · {^{n+1} }v(x) · \mathrm{d}x$ $= \left[ {^0}u(x)·{^n}v(x) \right]{x_α}^{x_β} - \int{x_α}^{x_β} {^1}u(x) ·{^n}v(x) · \mathrm{d} x$
  $= \left[ ^{0}u(x)·{^n}v(x) - {^1}u(x) · {^{n-1} }v(x) \right]{x_α}^{x_β} + \int\limits{x_α}^{x_β} {^2}u(x) ·{^{n-1} }v(x) ·\mathrm{d} x$
  $= \left[ \sum\limits_{i=0}^{n} (-1)^{n} ·{^i}u(x) ·{^{n-i} }v(x) \right]{x_α}^{x_β} + (-1)^{n+1} \int\limits{x_α}^{x_β} {^{n+1} }u(x) · {^0}v(x) ·\mathrm{d}x$

典例:应用换元积分法,计算$\int \tan x · \mathrm{d}x$。

$\int_x \tan x · \mathrm{d}x = \int_x \dfrac{\sin x}{\cos x} · \mathrm{d}x = \int_x \dfrac{-1}{\cos x} · \mathrm{d} \cos x \mathop{====}\limits^{t=\cos x} \int_t \dfrac{-1}{t} · \mathrm{d} t = -\ln t + \mathrm{Con} = -\ln \cos x + \mathrm{Con}$

典例:应用换元积分法,计算$\int \dfrac{1}{x^2 - a^2} · \mathrm{d}x$。

$\int_x \dfrac{1}{x^2 - a^2} · \mathrm{d}x = \dfrac{1}{2 · a} · \int_x \left[ \dfrac{1}{x - a} - \dfrac{1}{x + a} \right] \mathrm{d} x = \dfrac{1}{2·a} · \left[ \int_x \dfrac{\mathrm{d}(x-a)}{x-a} - \int_x \dfrac{\mathrm{d}(x+a)}{x + a} \right] \mathop{===}\limits_{s=x+a}^{t=x-a} \dfrac{1}{2·a} · \int_t \dfrac{\mathrm{d}t}{t} - \dfrac{1}{2·a} · \int_s \dfrac{\mathrm{d}s}{s} = \dfrac{1}{2·a} · \left[ \ln t - \ln s \right] + \mathrm{Con}= \dfrac{1}{2·a}· \ln \left \dfrac{x-a}{x+a} \right + \mathrm{Con}$

典例:应用换元积分法,计算$\int \dfrac{1}{\sqrt{x^2 - a^2} } · \mathrm{d}x$。已知$\int \dfrac{1}{\sqrt{x^2 - 1} } · \mathrm{d}x = {‘}\cosh x + \mathrm{Con}$。

$\int \dfrac{1}{\sqrt{x^2 - a^2} } · \mathrm{d}x = \int \dfrac{1}{\sqrt{\frac{x^2}{a^2} - 1^2} } · \mathrm{d} \dfrac{x}{a} \mathop{==}\limits^{t=\frac{x}{a} } {‘}\cosh t + \mathrm{Con}_0 = {‘}\cosh \dfrac{x}{a} + \mathrm{Con}_0 = \ln\left( \dfrac{x}{a} + \sqrt{\dfrac{x^2}{a^2} - 1} \right) + \mathrm{Con}_0 = \ln \left( x + \sqrt{x^2 + a^2} \right) + \mathrm{Con}$

典例:应用分部积分法$x$降幂型式,计算$\int x· ә^{-x} · \mathrm{d}x$。

$\int x· ә^{-x} · \mathrm{d}x = -\int x · \mathrm{d} ә^{-x} = -x · ә^{-x} + \intә^{-x} · \mathrm{d} x = -x·ә^{-x} + \int \mathrm{d}ә^{-x} = -x·ә^{-x} + ә^{-x} + \mathrm{Con}$

典例:应用分部积分法$x$升幂型式,计算$\int x · \ln x · \mathrm{d}x$。

$\int x · \ln x · \mathrm{d}x = \dfrac{1}{2} · \int \ln x · \mathrm{d} x^2 = \dfrac{1}{2} · \left[ x^2 · \ln x - \int x^2 · \mathrm{d}\ln x \right] \mathop{=====}\limits^{\mathrm{d}\ln x = \frac{1}{x}·\mathrm{d}x} \dfrac{1}{2} · x^2 · \ln x - \dfrac{1}{2} · \int x · \mathrm{d} x = \dfrac{1}{2} · x^2 · \ln x - \dfrac{1}{4} ·x^2 + \mathrm{Con}$

典例:应用分部积分法循环型式,计算$\int \sqrt{x^2 - a^2}·\mathrm{d}x$。已知$\int \dfrac{1}{\sqrt{x^2 - a^2} } · \mathrm{d}x = {‘}\cosh \dfrac{x}{a} + \mathrm{Con}$。

$\int \sqrt{x^2 - a^2}·\mathrm{d}x = x · \sqrt{x^2 - a^2} - \int x · \mathrm{d} \sqrt{x^2 - a^2} = x · \sqrt{x^2 + a^2} - \int x · \dfrac{2·x}{2\sqrt{x^2-a^2} } \mathrm{d}x = x· \sqrt{x^2 - a^2} - \int \dfrac{(x^2 - a^2)+a^2}{\sqrt{x^2 - a^2} } · \mathrm{d}x = x·\sqrt{x^2-a^2} - \int \sqrt{x^2-a^2} · \mathrm{d}x - a^2 · \int \dfrac{1}{\sqrt{x^2 - a^2} } · \mathrm{d}x$

$\int \sqrt{x^2-a^2} = \dfrac{x·\sqrt{x^2-a^2} }{2} - \dfrac{a^2}{2} · {‘}\cosh \dfrac{x}{a} + \mathrm{Con}$

典例:应用分部积分法递推型式,计算$\int \cos^n x · \mathrm{d}x$。

$\int \cos^n x · \mathrm{d}x = \int \cos^{n-1} x · \mathrm{d} \sin x = \cos^{n-1} x · \sin x - \int \sin x · \mathrm{d} \cos^{n-1} x = \cos^{n-1} x · \sin x + (n - 1) · \int \cos^{n-2}x · \sin^2 x ·\mathrm{d} x \mathop{=======}\limits^{\sin^2 x = 1 - \cos^2 x} \cos^{n-1} x · \sin x +(n-1) · \int [\cos^{n-2} x - \cos^n x]·\mathrm{d}x$

$\int \cos^{n} x · \mathrm{d} x = \dfrac{1}{n} · \cos^{n-1}x· \sin x + \dfrac{n - 1}{n} · \int \cos^{n-2}x·\mathrm{d}x$

若函数$f(x)$为奇函数且定积分存在,则$\int\limits_{-x_γ}^{+x_γ} f(x) · \mathrm{d}x = 0$。

若函数$f(x)$为偶函数且定积分存在,则$\int\limits_{-x_γ}^{+x_γ} f(x) · \mathrm{d} x = 2 · \int\limits_{0}^{+x_γ} f(x) · \mathrm{d}x$。

若函数$f(x)$为周期函数且定积分存在,则$\int\limits_{x_γ}^{x_γ+T} f(x) · \mathrm{d} x = \int\limits_{0}^{T} f(x) · \mathrm{d} x$。

$⇓$ $\int\limits_{-x_γ}^{+x_γ} f(x) · \mathrm{d} x = \int\limits_{-x_γ}^{0} f(x) · \mathrm{d} x + \int\limits_{0}^{+x_γ} f(x) · \mathrm{d} x \mathop{===}\limits^{x=-t} \int\limits_{x_γ}^{0} f(-t) · \mathrm{d}(-t) + \int\limits_{0}^{x_γ} f(x) · \mathrm{d} x \mathop{======}\limits^{f(-t)=-f(t)} \int\limits_{0}^{x_γ} -f(t) · \mathrm{d}t + \int\limits_{0}^{x_γ} f(x)·\mathrm{d}x$
   
$⇓$ $\int\limits_{-x_γ}^{+x_γ} f(x) · \mathrm{d} x = \int\limits_{-x_γ}^{0} f(x)· \mathrm{d}x + \int\limits_{0}^{+x_γ} f(x) · \mathrm{d} x \mathop{===}\limits^{x=-t} \int\limits_{x_γ}^{0} f(-t)·\mathrm{d}(-t) + \int\limits_{0}^{x_γ} f(x) · \mathrm{d} x \mathop{======}\limits^{f(-t)=+f(t)} \int\limits_{0}^{x_γ} +f(t) · \mathrm{d}t + \int\limits_{0}^{x_γ} f(x)·\mathrm{d}x$
   
$⇓$ $\int\limits_{T}^{x_γ+T} f(x) · \mathrm{d} x \mathop{====}\limits^{x=t+T} \int\limits_{0}^{x_γ} f(t+T) · \mathrm{d}(t+T) \mathop{======}\limits^{f(t+T)=f(t)} \int\limits_{0}^{x_γ} f(t) · \mathrm{d}t$
$⇓$ $\int\limits_{x_γ}^{x_γ+T} f(x) · \mathrm{d} x = \int\limits_{x_γ}^{0} f(x) · \mathrm{d}x + \int\limits_{0}^{T} f(x) · \mathrm{d}x + \int\limits_{T}^{x_γ+T} f(x) · \mathrm{d}x = -\int\limits_{0}^{x_γ} f(x) · \mathrm{d}x + \int\limits_{0}^{T} f(x) · \mathrm{d}x + \int\limits_{0}^{x_γ} f(x) · \mathrm{d}x = \int\limits_{0}^{T} f(x) · \mathrm{d}x$

积分中值定理

错位积和公式。

$\inf F (x) · g^{↘0} (x_α) ≤ \int\limits{x_α}^{x} f (θ_{i}) · g^{↘0} (θ{i}) · Δx_i ≤ \sup F (x) · g^{↘_0} (x_α)$

$\inf F (x) · g^{↗^0} (x_α) ≥ \int\limits_{x_α}^{x} f (θ_{i}) · g^{↗^0} (θ_{i}) · Δx_i ≥ \sup F (x) · g^{↗^0} (x_α)$

$\inf F (x) · g^{↗0} (x_β) ≤ \int\limits{x}^{x_β} f (θ_{i}) · g^{↗0} (θ{i}) · Δx_i ≤ \sup F (x) · g^{↗_0} (x_β)$

$\inf F (x) · g^{↘^0} (x_β) ≥ \int\limits_{x}^{x_β} f (θ_{i}) · g^{↘^0} (θ_{i}) · Δx_i ≥ \sup F (x) · g^{↘^0} (x_β)$

$⇓$ $F(x_{n+1}) \mathop{≡≡≡≡≡≡}\limits_{x_i≤θ_i≤x_{i+1} }^{x_α≤θ_i≤x} \sum\limits_{i=0}^{n} f(θ_i) · Δx_i$   $\int\limits_{x_α}^{x} f (x) \mathrm{d} x = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} F(x_{n+1})$
$⇓$ $\sum\limits_{i=n}^{m} f (θ_{i}) · g (θ_{i}) · Δx_i = \sum\limits_{i=n}^{m} [ F (x_{i+1}) - F (x_{i}) ] · g (θ_{i}) = F (x_{m+1}) · g (θ_{m}) + \sum\limits_{i=n}^{m-1} F (x_{i+1}) · g (θ_{i}) - \sum\limits_{i=n}^{m-1} F (x_{i+1}) · g (θ_{i+1}) - F (x_{n}) · g (θ_{n})$    
$⇓$ $\sum\limits_{i=0}^{m} f (θ_{i}) · g (θ_{i}) · Δx_i = [ F (x_{m+1}) · g (θ_{m}) - \rlap{≡≡≡≡≡≡≡}{F (x_{0}) · g (θ_{0})} ] + \sum\limits_{i=0}^{m-1} F (x_{i+1}) · [ g (θ_{i}) - g (θ_{i+1}) ]$ $⇐$ $F (x_{0}) ≡ 0$
$⇓$ $\sum\limits_{i=0}^{m} f (θ_i) · g^{↘0} (θ_i) · Δx_i ≤ \sup F (x) · g^{↘_0} (θ{m}) + \sup F (x) · \sum\limits_{i=0}^{m-1} [ g^{↘0} (θ{i}) - g^{↘0} (θ{i+1}) ] = \sup F (x) · g^{↘_0} (θ_0)$ $⇐$ $g^{↘0} (θ{i}) ≥ g^{↘0} (θ{i+1}) ≥ 0$
$⇓$ $\sum\limits_{i=0}^{m} f (θ_i) · g^{↘0} (θ_i) · Δx_i ≥ \inf F (x) · g^{↘_0} (θ{m}) + \inf F (x) · \sum\limits_{i=0}^{m-1} [ g^{↘0} (θ{i}) - g^{↘0} (θ{i+1}) ] = \inf F (x) · g^{↘0} (θ{0})$ $⇐$ $g^{↘0} (θ{i}) ≥ g^{↘0} (θ{i+1}) ≥ 0$
$⇓$ $\inf F (x) · g^{↘0} (x_α) ≤ \int\limits{x_α}^{x} f (θ_{i}) · g^{↘0} (θ{i}) · Δx_i ≡ \lim\limits_{m⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{m} f (θ_{i}) · g^{↘0} (θ{i}) · Δx_i ≤ \sup F (x) · g^{↘_0} (x_α)$ $⇐$ $x_0 = x_α$
$⇓$ $\inf F (x) · g^{↗^0} (x_α) ≥ \int\limits_{x_α}^{x} f (θ_{i}) · g^{↗^0} (θ_{i}) · Δx_i ≡ \lim\limits_{m⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{m} f (θ_{i}) · g^{↗^0} (θ_{i}) · Δx_i ≥ \sup F (x) · g^{↗^0} (x_α)$ $⇐$ $g^{↗^0} (x) ≡ (-1) · h^{↘_0} (x)$
       
$⇓$ $F(x_{n+1}) \mathop{≡≡≡≡≡≡}\limits_{x_i≤θ_i≤x_{i+1} }^{x≤θ_i≤x_β} \sum\limits_{i=0}^{n} f(θ_i) · Δx_i$   $\int\limits_{x}^{x_β} f (x) \mathrm{d} x = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} F(x_{n+1})$
$⇓$ $\sum\limits_{i=n}^{m} f (θ_{i}) · g (θ_{i}) · Δx_i = \sum\limits_{i=n}^{m} [ F (x_{i+1}) - F (x_{i}) ] · g (θ_{i}) = F (x_{m+1}) · g (θ_{m}) + \sum\limits_{i=n}^{m-1} F (x_{i+1}) · g (θ_{i}) - \sum\limits_{i=n}^{m-1} F (x_{i+1}) · g (θ_{i+1}) - F (x_{n}) · g (θ_{n})$    
$⇓$ $\sum\limits_{i=0}^{m} f (θ_{i}) · g (θ_{i}) · Δx_i = [ F (x_{m+1}) · g (θ_{m}) - \rlap{≡≡≡≡≡≡≡}{F (x_{0}) · g (θ_{0})} ] + \sum\limits_{i=0}^{m-1} F (x_{i+1}) · [ g (θ_{i}) - g (θ_{i+1}) ]$ $⇐$ $F (x_{0}) ≡ 0$
$⇓$ $\sum\limits_{i=0}^{m} f (θ_i) · g^{↗0} (θ_i) · Δx_i ≤ \sup F (x) · g^{↗_0} (θ{m}) + \sup F (x) · \sum\limits_{i=0}^{m-1} [ g^{↗0} (θ{i}) - g^{↗0} (θ{i+1}) ] = \sup F (x) · g^{↗_0} (θ_0)$ $⇐$ $0 ≤ g^{↗0} (θ{i+1}) ≤ g^{↗0} (θ{i})$
$⇓$ $\sum\limits_{i=0}^{m} f (θ_i) · g^{↗0} (θ_i) · Δx_i ≥ \inf F (x) · g^{↗_0} (θ{m}) + \inf F (x) · \sum\limits_{i=0}^{m-1} [ g^{↗0} (θ{i}) - g^{↗0} (θ{i+1}) ] = \inf F (x) · g^{↗_0} (θ_0)$ $⇐$ $0 ≤ g^{↗0} (θ{i+1}) ≤ g^{↗0} (θ{i})$
$⇓$ $\inf F (x) · g^{↗0} (X_1) ≤ \int\limits{x}^{x_β} f (θ_{i}) · g^{↗0} (θ{i}) · Δx_i ≡ \lim\limits_{m⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{m} f (θ_i) · g^{↗_0} (θ_i) · Δx_i ≤ \sup F (x) · g^{↗_0} (X_1)$ $⇐$ $x_0 = x_β$
$⇓$ $\inf F (x) · g^{↘^0} (X_1) ≥ \int\limits_{x}^{x_β} f (θ_{i}) · g^{↘^0} (θ_{i}) · Δx_i ≡ \lim\limits_{m⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{m} f (θ_i) · g^{↘^0} (θ_i) · Δx_i ≥ \sup F (x) · g^{↘^0} (X_1)$ $⇐$ $g^{↘^0} (x) ≡ (-1) · h^{↗_0} (x)$

若函数$f (x)$在区间$[x_α,x_β]$上连续,函数$g (x)$在区间$[x_α,x_β]$上连续,则必至少存在一点$θ∈(x_α, x_β)$使得成立。[积分第一中值定理]

$\int\limits_{x_α}^{x_β} f (x) · g (x) \mathrm{d} x = \left. f (x) \right {∃θ∈(x_α, x_β)} · \int\limits{x_α}^{x_β} g (x) \mathrm{d} x$
$\int\limits_{x_α}^{x_β} f (x) \mathrm{d} x = \left. f (x) \right _{∃θ∈[x_α,x_β]} · (x_β-x_α)$
$⇓$ $F (x) ≡ \int\limits_{x_α}^{x} f (x) · g (x) \mathrm{d} x$ $⇒$ $F (x_α) = 0; F (x_β) = \int\limits_{x_α}^{x_β} f (x) · g (x) \mathrm{d} x$      
$⇓$ $G (x) ≡ \int\limits_{x_α}^{x} g (x) \mathrm{d} x$ $⇒$ $G (x_α) = 0; G (x_β) = \int\limits_{x_α}^{x_β} g (x) \mathrm{d} x$      
$⇓$ $\dfrac{\int\limits_{x_α}^{x_β} f (x) · g (x) \mathrm{d} x}{\int\limits_{x_α}^{x_β} g (x) \mathrm{d} x} = \dfrac{F (x_β) - F (x_α)}{G (x_β) - G (x_α)} = \left. \dfrac{\mathrm{d} F (x)}{\mathrm{d} G (x)} \right _{∃θ∈[x_α, x_β]} = \left. \dfrac{f (x) · g (x)}{g (x)} \right _{∃θ∈[x_α, x_β]} = \left. f (x) \right _{∃θ∈[x_α, x_β]}$    
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g (x) \mathrm{d} x = \left. f (x) \right {∃θ∈[x_α, x_β]} · \int\limits{x_α}^{x_β} g (x) \mathrm{d} x$        
$⇓$ $\int\limits_{x_α}^{x_β} f (x) \mathrm{d} x = \left. f (x) \right _{∃θ∈[x_α, x_β]} · (x_β - x_α)$ $⇐$ $g (x) ≡ 1$    

若函数$f^{⤨} (x)$在区间$[x_α, x_β]$上连续且单调,函数$g (x)$在区间$[x_α, x_β]$上连续,则必至少存在一点$θ∈[x_α,x_β]$使得成立。[积分第二中值定理]

$\int\limits_{x_α}^{x_β} f (x) · g^{↘0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↘_0} (x_α) · \int\limits{x_α}^{θ} f (x) \mathrm{d} x$

$\int\limits_{x_α}^{x_β} f (x) · g^{↗^0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↗^0} (x_α) · \int\limits_{x_α}^{θ} f (x) \mathrm{d} x$

$\int\limits_{x_α}^{x_β} f (x) · g^{↗0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↗_0} (x_β) · \int\limits{θ}^{x_β} f (x) \mathrm{d} x$

$\int\limits_{x_α}^{x_β} f (x) · g^{↘^0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↘^0} (x_β) · \int\limits_{θ}^{x_β} f (x) \mathrm{d} x$

$\int\limits_{x_α}^{x_β} f (x) · g^{⤨} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{⤨} (x_α) · \int\limits_{x_α}^{θ} f (x) \mathrm{d} x + g^{⤨} (x_β) · \int\limits_{θ}^{x_β} f (x) \mathrm{d} x$

$⇓$     $∀F (x)∈[\inf F (x), \sup F (x)];∃θ∈[x_α,x_β]; F (x) = F (θ)$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g^{↘0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↘_0} (x_α) · \int\limits{x_α}^{θ} f (x) \mathrm{d} x$ $⇐$ $\inf F (x) · g^{↘0} (x_α) ≤ \int\limits{x_α}^{x} f (θ_{i}) · g^{↘0} (θ{i}) · Δx_i ≤ \sup F (x) · g^{↘_0} (x_α)$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g^{↗^0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↗^0} (x_α) · \int\limits_{x_α}^{θ} f (x) \mathrm{d} x$ $⇐$ $\inf F (x) · g^{↗^0} (x_α) ≥ \int\limits_{x_α}^{x} f (θ_{i}) · g^{↗^0} (θ_{i}) · Δx_i ≥ \sup F (x) · g^{↗^0} (x_α)$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g^{↗0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↗_0} (x_β) · \int\limits{θ}^{x_β} f (x) \mathrm{d} x$ $⇐$ $\inf F (x) · g^{↗0} (x_β) ≤ \int\limits{x}^{x_β} f (θ_{i}) · g^{↗0} (θ{i}) · Δx_i ≤ \sup F (x) · g^{↗_0} (x_β)$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g^{↘^0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{↘^0} (x_β) · \int\limits_{θ}^{x_β} f (x) \mathrm{d} x$ $⇐$ $\inf F (x) · g^{↘^0} (x_β) ≥ \int\limits_{x}^{x_β} f (θ_{i}) · g^{↘^0} (θ_{i}) · Δx_i ≥ \sup F (x) · g^{↘^0} (x_β)$
       
$⇓$ $h^{↘_0} (x) ≡ \dfrac{g^{⤨} (x) - g^{⤨} (x_β)}{g^{⤨} (x_α) - g^{⤨} (x_β)}$ $⇒$ $\int\limits_{x_α}^{x_β} h^{↘0} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} h^{↘_0} (x_α) · \int\limits{x_α}^{θ} f (x) \mathrm{d} x$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · \dfrac{g^{⤨} (x) - g^{⤨} (x_β)}{g^{⤨} (x_α) - g^{⤨} (x_β)} \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} 1 · \int\limits_{X_0}^{θ} f (x) \mathrm{d} x$ $⇐$ $h^{↘_0} (x_α) = 1$
$⇓$ $\int\limits_{x_α}^{x_β} f (x) · g^{⤨} (x) \mathrm{d} x \mathop{=====}\limits^{∃θ∈[x_α,x_β]} g^{⤨} (x_α) · \int\limits_{x_α}^{θ} f (x) \mathrm{d} x + g^{⤨} (x_β) · \int\limits_{θ}^{x_β} f (x) \mathrm{d} x$