特殊函数
更新: 作者: 资源:
绝对函数$\mathrm{abs}(x)$
| $\mathrm{abs}(x) = | x | = \mathop{-x}\limits_{x<0};\mathop{0}\limits_{x=0};\mathop{+x}\limits_{x>0}$ |
| $\dfrac{\Lambda \mathrm{abs}(0)}{\lambda x} = \lim\limits_{x⇝0} \dfrac{\mathrm{abs}(x) - \mathrm{abs}(0)}{x - 0} = \lim\limits_{x⇝0} \dfrac{ | x | }{x}$ |
| $\dfrac{\Lambda \mathrm{abs}(x)}{\lambda x} = \mathop{-1}\limits_{x<0};\mathop{ | x | ·x^{-1} }\limits_{x=0};\mathop{+1}\limits_{x>0} = \dfrac{ | x | }{x}$ |
特征函数$\mathcal{X}_{I} (x)$
$\mathcal{X}{I} (x) = \mathop{1}\limits{x∈I};\mathop{0}\limits_{x∉I}$
有理函数$\mathcal{Q} (x)$
$\mathcal{Q} (x) = \mathop{1}\limits_{x∈ℚ}; \mathop{0}\limits_{x∉ℚ}$
函数$\mathcal{Q} (x)$在区间$ℝ$内，处处不连续，处处不可导。
| $⇓$ | $∀x_0∈ℝ;∃ε=1;∀δ>0; \sup\limits_{u,v∈\mathrm{U}(x_0,δ)} \mathop{ | f(u) - f(v) | }\limits_{u∈ℚ,v∉ℚ} = ε$ |
|---|---|---|---|
| $⇓$ | $\lim\limits_{δ⇝0} w^f (x_0,δ) \not⇝ 0$ |
函数$\mathcal{Q} (x)$在区间$[0, 1]$上有确界，但其定积分不存在。
$\lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \inf\limits_{x∈[x_i, x_i+1]} \mathcal{Q} (x) · Δx_i = \lim\limits_{n⇝∞⁺} \sum\limits_{i=0}^{n} 0 · Δx_i ⇝ 0 ≠ 1 ⇜ \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} 1 · Δx_i = \lim\limits_{n⇝∞⁺}^{Δx_i⇝0} \sum\limits_{i=0}^{n} \sup\limits_{x∈[x_i, x_{i+1}]} \mathcal{Q} (x) · Δx_i$
函数$\mathcal{Q}(x)$不存在原函数$F(x)$，使得$\dfrac{\mathrm{d} F(x)}{\mathrm{d}x} = \mathcal{Q}(x)$。
| $Ⅎ\dfrac{1}{2}∈[0,1]=[{^1}F(\sqrt{2}),{^1}F(0)];∃θ∈[0,\sqrt{2}]; {^1}F(θ) = \mathcal{Q}(θ) = \dfrac{1}{2}$ | $\mathrm{False}$ |
|---|---|
实数函数$\mathcal{R} (x)$
| $\mathcal{R} (x) \mathop{======}\limits_{m∈ℤ;n∈ℤ^+}^{1 = \gcd ( | m | , | n | )} \mathop{\dfrac{1}{n} }\limits_{x=\frac{m}{n} }; \mathop{0}\limits_{x≠\frac{m}{n} }$ |
函数$\mathcal{R} (x)$在区间$ℝ$内，任何有理点处不连续，任何无理点处连续。 | $⇓$ | $∀x_0∉ℚ;∀ε>0;∃δ>0;∀x∈ℝ; \left[\left| \mathcal{R}(x) - \mathcal{R}(x_0) \right| > ε\right] ⇒ \left[ x ∈ \left\lbrace ∀m_j∈ℤ;∀n_i∈ℕ; \dfrac{m_j}{n_i} : \dfrac{1}{n_i} > ε \right\rbrace \right]$ | $¬Q ⇒ ¬P$ | | :–: | :———————————————————– | :——– | | $⇓$ | $∀x_0∉ℚ;∀ε>0;∃δ>0;∀x∈ℝ; \left[ |x - x_0| < \min\left\lbrace ∀m_j∈ℤ;∀n_i∈ℕ; \left| \dfrac{m_j}{n_i} - x_0 \right|: \dfrac{1}{n_i} > ε \right\rbrace = δ \right] ⇒ \left[ \left| \mathcal{R}(x) - \mathcal{R}(x_0) \right| = \left| \mathop{\dfrac{1}{n} }\limits_{x=\frac{m}{n} };\mathop{0}\limits_{x≠\frac{m}{n} } \right| ≤ ε \right]$ | $P ⇒ Q$ | | $⇓$ | $\lim\limits_{x⇝x_0≠\frac{p}{q} } \mathcal{R} (x) ⇝ \mathcal{R} (x_0)$ | | | | | | | $⇓$ | $∃x_0=\dfrac{p}{q}∈ℚ;∃ε>0;∀δ>0; \sup\limits_{x∈\mathrm{U}(x_0,δ)} \left| \mathcal{R}(x) - \mathcal{R}(x_0) \right| = \sup\limits_{x∈\mathrm{U}(x_0,δ)} \left| \mathop{\dfrac{1}{n} }\limits_{x=\frac{m}{n} };\mathop{0}\limits_{x≠\frac{m}{n} } - \dfrac{1}{q} \right| ≥ \dfrac{1}{q}$ | | | $⇓$ | $\lim\limits_{x⇝x_0=\frac{p}{q} } \mathcal{R} (x) \not⇝ \mathcal{R} (x_0)$ | |
必存在函数$f (x)$在区间$ℝ$内，任何有理点处连续，任何无理点处连续。例如$f (x) ≡ x$。
必存在函数$f (x)$在区间$ℝ$内，任何有理点处不连续，任何无理点处不连续。例如$f (x) ≡ \mathcal{Q}(x)$。
必存在函数$f (x)$在区间$ℝ$内，任何有理点处不连续，任何无理点处连续。例如$f (x) ≡ \mathcal{R}(x)$。
$∃f(x); \left[ ∀q_0∈ℚ;∃ε>0;∀δ>0; w^{f}(q_0,δ) > ε \right] ∧ \left[ ∀r_0∉ℚ;∀ε>0;∃δ>0; w^{f}(r_0,δ) ≤ ε \right]$
$∃f(x);∀r∈ℝ; [r∉ℚ] ⇔ \left[ ∀ε>0;∃δ>0; w^{f}(r,δ) ≤ ε \right]$
不存在函数$f (x)$在区间$ℝ$内，任何有理点处连续，任何无理点处不连续。注意：有理数的间隙由无理数填充，反之不对。
$¬∃f(x); \left[ ∀q_0∈ℚ;∀ε>0;∃q_1>0; w^{f}(q_0,q_1) ≤ ε \right] ∧ \left[ ∀r_0∉ℚ;∃ε>0;∀δ>0; w^{f}(r_0,δ) > ε \right]$
$¬∃f(x);∀q∈ℝ; [q∈ℚ] ⇔ \left[ ∀ε>0;∃δ>0; w^{f}(q,δ) ≤ ε \right]$
| $⇓$ | $∃f(x); \left[ ∀q_0∈ℚ;∀ε_q>0;∃q_1>0; w^{f}(q_0,q_1) ≤ ε_q \right] ∧ \left[ ∀r_0∉ℚ;∃ε_r>0;∀δ>0; w^{f}(r_0,δ) > ε_r \right]$ | $P$ | ||||
|---|---|---|---|---|---|---|
| $⇓$ | $∃f(x); ∀q_0∈ℚ;∀q_1∈ℚ; ∃r_0∉ℚ,r_0∈[q_0,q_1];∃ε_r=\mathrm{Con}>0;∀δ>0; w^{f}(r_0,δ) > ε_r = \mathrm{Con} > 0$ | |||||
| $⇓$ | $∃f(x); ∀q_0∈ℚ;∀ε_q>0;∃q_1>0; ∃r_0∉ℚ,r_0∈[q_0,q_1];∃ε_r=\mathrm{Con}>0;∀δ=\min\lbrace | q_0-r_0 | , | q_1-r_0 | \rbrace; 0 < \mathrm{Con} = ε_r < w^{f}(r_0,δ) ≤ w^{f}(q_0,q_1) ≤ ε_q$ | |
| $⇓$ | $∃f(x); ∀q_0∈ℚ;∀ε_q>0;∃q_1>0; \mathrm{Con} ≤ w^{f}(q_0,q_1) ≤ ε_q$ | $Q = \mathrm{False}$ | ||||
| $⇓$ | $¬∃f(x); \left[ ∀q_0∈ℚ;∀ε>0;∃q_1>0; w^{f}(q_0,q_1) ≤ ε \right] ∧ \left[ ∀r_0∉ℚ;∃ε>0;∀δ>0; w^{f}(r_0,δ) > ε \right]$ | $¬Q ∧ (P → Q) ⇒ ¬P$ | ||||
| $⇓$ | $¬∃f(x); \left[ ∀q∈ℝ; [q∈ℚ] ⇒ [∀ε>0;∃δ>0; w^{f}(q,δ) ≤ ε] \right] ∧ \left[ ∀q∈ℝ; [q∉ℚ] ⇒ [∃ε>0;∀δ>0; w^{f}(q,δ) > ε] \right]$ | |||||
| $⇓$ | $¬∃f(x);∀q∈ℝ; [q∈ℚ] ⇔ \left[ ∀ε>0;∃δ>0; w^{f}(q,δ) ≤ ε \right]$ |
分形函数$\mathcal{F} (x)$
$\lbrace (X_{i,j} - I_{i}, X_{i,j} + I_{i}) \rbrace \mathop{≡≡≡≡≡}\limits_{I_{i}=\frac{1}{2^{2·i+3} } }^{X_{i,j}=\frac{2·j+1}{2^{i+1} } } \bigcup\limits_{i=0}^{+∞} \bigcup\limits_{j=0}^{2^i-1} \left( \dfrac{2 · j + 1}{2^{i + 1} } - \dfrac{1}{2^{2 · i + 3} }, \dfrac{2 · j + 1}{2^{i + 1} } + \dfrac{1}{2^{2 · i + 3} } \right) = \bigcup\limits_{i=1}^{+∞} \bigcup\limits_{j=1}^{2^{i-1} } \left( \dfrac{2·j-1}{2^i} - \dfrac{1}{2^{2·i+1} }, \dfrac{2·j-1}{2^i} + \dfrac{1}{2^{2·i+1} } \right)$
| $ | \lbrace (X_{i,j} - I_{i}, X_{i,j} + I_{i}) \rbrace | = \sum\limits_{i=0}^{+∞} \sum\limits_{j=0}^{2^i - 1} 2 · I_i = \sum\limits_{i=0}^{+∞} \sum\limits_{j=0}^{2^i-1} \dfrac{2}{2^{2·i+3} } = \sum\limits_{i=0}^{+∞} 2^i · \dfrac{2}{2^{2·i+3} } = \sum\limits_{i=0}^{+∞} \dfrac{1}{2^{i+2} } = \dfrac{\frac{1}{4} }{1 - \frac{1}{2} } = \dfrac{1}{2}$ |
| $I_0 = \dfrac{1}{2^3}$ | $\left( \dfrac{1}{2^1} - \dfrac{1}{2^3}, \dfrac{1}{2^1} + \dfrac{1}{2^3} \right)$ |
|---|---|
| $I_1 = \dfrac{1}{2^5}$ | $\left( \dfrac{1}{2^2} - \dfrac{1}{2^5}, \dfrac{1}{2^2} + \dfrac{1}{2^5} \right) ∪ \left( \dfrac{3}{2^2} - \dfrac{1}{2^5}, \dfrac{3}{2^2} + \dfrac{1}{2^5} \right)$ |
| $I_2 = \dfrac{1}{2^7}$ | $\left( \dfrac{1}{2^3} - \dfrac{1}{2^7}, \dfrac{1}{2^3} + \dfrac{1}{2^7} \right) ∪ \left( \dfrac{3}{2^3} - \dfrac{1}{2^7}, \dfrac{3}{2^3} + \dfrac{1}{2^7} \right) ∪ \left( \dfrac{5}{2^3} - \dfrac{1}{2^7}, \dfrac{5}{2^3} + \dfrac{1}{2^7} \right) ∪ \left( \dfrac{7}{2^3} - \dfrac{1}{2^7}, \dfrac{7}{2^3} - \dfrac{1}{2^7} \right)$ |
| $\mathcal{F} (x) ≡ \mathop{I_i - \left | x - X_{i,j} \right | }\limits_{ | x - X_{i,j} | < I_{i} }; \mathop{0}\limits_{ | x - X_{i,j} | ≥ I_{i} }$ |
函数$\mathcal{F} (x)$在区间$[0, 1]$内连续，在以点$X_{i,j}$为中点且以$I_i$为半径的区间内呈三角形状，其三角形状区间的可测度长度为$\dfrac{1}{2}$。
符号函数$\mathrm{sgn} (x)$
| $\mathrm{sgn} (x) = \mathop{-1}\limits_{x<0};\mathop{0}\limits_{x=0};\mathop{+1}\limits_{x>0} = \mathop{0}\limits_{x=0};\mathop{x· | x | ^{-1} }\limits_{x≠0}$ |
$\mathrm{sgn} (\mathcal{R} (x)) = \mathcal{Q} (x)$
| $\dfrac{\mathrm{d} \mathrm{sgn}(0)}{\mathrm{d} x} = \lim\limits_{x⇝0} \dfrac{\mathrm{sgn}(x) - \mathrm{sgn}(0)}{x - 0} = \lim\limits_{x⇝0} \dfrac{1}{ | x | }$ |
| $\dfrac{\mathrm{d} \mathrm{sgn}(x)}{\mathrm{d} x} = \mathop{\dfrac{0}{x} }\limits_{x<0};\mathop{\dfrac{-1}{x} }\limits_{x=0^{-} };\mathop{\dfrac{+1}{x} }\limits_{x=0^{+} };\mathop{\dfrac{0}{x} }\limits_{x>0} = \mathop{\dfrac{1}{ | x | } }\limits_{x=0};\mathop{\dfrac{0}{x} }\limits_{x≠0}$ |
$\lim\limits_{x⇝0} \dfrac{\mathrm{d} \mathrm{sgn}(x)}{\mathrm{d} x} ⇝ 0 ≠ \dfrac{\mathrm{d} \mathrm{sgn}(0)}{\mathrm{d} x}$
振荡函数$\mathcal{W}_{α}(x)$
$\mathcal{W}{α}(x) = \mathop{0}\limits{x=0};\mathop{x^{α} · \sin \dfrac{1}{x} }\limits_{x≠0}$
$\dfrac{\mathrm{d} \mathcal{W}{α}(0)}{\mathrm{d} x} = \lim\limits{x⇝0} \dfrac{\mathcal{W}{α}(x) - \mathcal{W}{α}(0)}{x - 0} = \lim\limits_{x⇝0} x^{α-1} · \sin \dfrac{1}{x}$
$\dfrac{\mathrm{d} \mathcal{W}{α}(x)}{\mathrm{d} x} = \mathop{\left[ x^{α-1} · \sin \dfrac{1}{x} \right]}\limits{x=0};\mathop{\left[ α · x^{α-1} · \sin \dfrac{1}{x} - x^{α-2} · \cos \dfrac{1}{x} \right]}\limits_{x≠0}$
当$α ≤ 0$时，$\mathcal{W}_{α}(x)$在点$x = 0$处振荡，因此其导数振荡，其导函数振荡，非一致连续。
当$0 < α ≤ 1$时，$\mathcal{W}_{α} (x)$在点$x = 0$处连续，其导数振荡，其导函数振荡，也一致连续。
当$1 < α ≤ 2$时，$\mathcal{W}_{α}(x)$在点$x = 0$处连续，其导数收敛，其导函数振荡，也一致连续。
当$2 < α$时，$\mathcal{W}{α}(x)$在点$x = 0$处连续，并且其导数收敛，其导函数收敛，也一致连续。 | $α < 0$ | $\lim\limits{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left| x^{α} · \sin \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝∞}^{x=[m·π+\frac{π}{2}]^{-1} } |x^{α}| \mathop{⇝}\limits_{n<0} ∞^{+}$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left| x^{α} · \cos \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝∞}^{x=[m·π]^{-1} } |x^{α}| \mathop{⇝}\limits_{n<0} ∞^{+}$ | $x = 0$处振荡 | | :———- | :———————————————————– | :———————————————————– | :———— | | $α = 0$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left|x^{α} · \sin \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝0}^{x=[m·π+\frac{π}{2}]^{-1} } \left|\sin \dfrac{1}{x}\right| = 1$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left|x^{α} · \cos \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝0}^{x=[m·π]^{-1} } \left|\cos \dfrac{1}{x}\right| = 1$ | $x = 0$处振荡 | | $0 < α$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left| x^{α} · \sin \dfrac{1}{x} \right| ≤ \lim\limits_{x⇝0}^{δ⇝0} |x^{α}| \mathop{⇝}\limits_{0<α} 0$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left| x^{α} · \cos \dfrac{1}{x} \right| ≤ \lim\limits_{x⇝0}^{δ⇝0} |x^{α}| \mathop{⇝}\limits_{0<α} 0$ | $x=0$处收敛 | | | | | | | $1 < α ≤ 2$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left|α · x^{α-1} · \sin \dfrac{1}{x} - x^{α-2} · \cos \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝∞}^{x=[m·π+\frac{π}{2}]^{-1} } |α·x^{α-1}| ⇝ \mathop{0}\limits_{1<α≤2}$ | $\lim\limits_{δ⇝0} \sup\limits_{x∈\mathrm{U}(0,δ)} \left|α · x^{α-1} · \sin \dfrac{1}{x} - x^{α-2} · \cos \dfrac{1}{x} \right| ≥ \lim\limits_{m⇝∞}^{x=[m·π]^{-1} } |x^{α-2}| ⇝ \mathop{∞^{+} }\limits_{1<α<2};\mathop{1}\limits_{α=2}$ | $x=0$处振荡 |
函数$f(x) ≡ \dfrac{\mathrm{d} \mathcal{W}{\frac{3}{2} }(x)}{\mathrm{d} x} = \mathop{0}\limits{x=0};\mathop{\left[ \dfrac{3}{2} · x^{\frac{1}{2} } · \sin \dfrac{1}{x} - \dfrac{1}{x^{\frac{1}{2} } } · \cos \dfrac{1}{x} \right]}\limits_{x≠0}$在点$x=0$处附近振荡且无界，但有原函数$F(x) ≡ \mathcal{W}{\frac{3}{2} }(x) = \mathop{0}\limits{x=0};\mathop{x^{\frac{3}{2} } · \sin \dfrac{1}{x} }\limits_{x≠0}$。
倒幂函数
$\mathcal{E}(x) ≡ \mathop{0}\limits_{x=0};\mathop{ә^{-\frac{1}{x} } }\limits_{x≠0}$
$\lim\limits_{x⇝0^{-} } ә^{-\frac{1}{x} } ⇝ ∞^{+}$
$\lim\limits_{x⇝0^{+} } ә^{-\frac{1}{x} } ⇝ 0^{+}$
脉冲函数
$\mathcal{P}(x) = \mathop{0}\limits_{};\mathop{+n^2·2^{n} · \left[ x - \left( n-\dfrac{1}{2^n · n} \right) \right]}\limits_{x∈\left[n-\frac{1}{2^{n}·n},n\right]};\mathop{-n^2·2^{n} · \left[ x - \left( n+\dfrac{1}{2^{n}·n} \right) \right]}\limits_{x∈\left[n,n+\frac{1}{2^{n}·n}\right]};0$
脉冲函数$\mathcal{P}(x)$为连续函数，在以点$n$处为中点且以$\dfrac{1}{2^n·n}$为半径的区间内呈三角形状，在点$n$处脉冲达到高度$n$。
脉冲函数$\mathcal{P}(x)$在点$n$处三角形的面积为$\dfrac{1}{2} · n · \dfrac{1}{2^{n}·n} · 2 = \dfrac{1}{2^n}$，所有三角形的面积和为$\int\limits_{0}^{∞^{+} } \mathcal{P}(x) · \mathrm{d}x = \sum\limits_{i=1}^{∞} \dfrac{1}{2^i} = 1$。
脉冲函数$\mathcal{P}(x)$在无穷大点处非一致连续。