ヘヴィサイドの階段関数

${\displaystyle {\begin{aligned}H(x)&={\begin{cases}0&(x<0)\\1&(x>0)\end{cases}}\\&={\dfrac {x+|x|}{2|x|}}\quad (x\neq 0)\end{aligned}}}$

${\displaystyle {\begin{aligned}U(x)&={\begin{cases}0&(x\leqq 0)\\1&(x>0)\end{cases}}\\&={\begin{cases}0&(x=0)\\{\dfrac {x+|x|}{2|x|}}&(x\neq 0)\end{cases}}\end{aligned}}}$

${\displaystyle {\begin{aligned}H_{c}(x)&={\begin{cases}0&(x<0)\\c&(x=0)\\1&(x>0)\end{cases}}\\&={\begin{cases}c&(x=0)\\{\dfrac {x+|x|}{2|x|}}&(x\neq 0)\end{cases}}\end{aligned}}}$

${\displaystyle H_{1}(x)=1-U(-x)=\lim _{t\to x-0}U(t)}$
${\displaystyle H_{0}(x)=U(x)}$
${\displaystyle H_{c}(x)=cH_{1}(x)+(1-c)H_{0}(x)}$
${\displaystyle H_{1/2}(x)={\frac {1+\operatorname {sgn}(x)}{2}}}$

${\displaystyle \int _{-\infty }^{x}\delta (t)dt:=\int _{-\infty }^{\infty }\chi _{(-\infty ,x]}(t)\delta (t)dt=\chi _{(-\infty ,x]}(0)}$

${\displaystyle \chi _{(-\infty ,x]}(0)={\begin{cases}0&(x<0)\\1&(x\geqq 0)\end{cases}}}$

${\displaystyle H_{1}(x)=\int _{-\infty }^{x}\delta (\xi )d\xi }$