Heaviside step function

In mathematics, physics, and engineering the Heaviside step function is the following function,

$H(x)={\begin{cases}1&\quad {\hbox{if}}\quad x>0\\{\frac {1}{2}}&\quad {\hbox{if}}\quad x=0\\0&\quad {\hbox{if}}\quad x<0\\\end{cases}}$

The function is named after the English mathematician Oliver Heaviside.

Derivative

Note that a block function B_Δ of width Δ and height 1/Δ can be given in terms of step functions (for positive Δ), namely

$B_{\Delta }(x)={\begin{cases}{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {0-0}{\Delta }}=0&\quad {\hbox{if}}\quad x<-\Delta /2\\{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {1-0}{\Delta }}={\frac {1}{\Delta }}&\quad {\hbox{if}}\quad -\Delta /2<x<\Delta /2\\{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}={\frac {1-1}{\Delta }}=0&\quad {\hbox{if}}\quad x>\Delta /2\\\end{cases}}$

Knowing this, the derivative of H follows easily

${\frac {dH(x)}{dx}}=\lim _{\Delta \rightarrow 0}{\frac {H(x+\Delta /2)-H(x-\Delta /2)}{\Delta }}=\lim _{\Delta \rightarrow 0}B_{\Delta }(x)=\delta (x),$

where δ(x) is the Dirac delta function, which may be defined as the block function in the limit of zero width, see this article.

The step function is a generalized function (a distribution). When H(x) is multiplied under the integral by the derivative of an arbitrary differentiable function f(x) that vanishes for plus/minus infinity, the result of the integral is minus the function value for x = 0,

$\int _{-\infty }^{\infty }H(x){\frac {df(x)}{dx}}\mathrm {d} x=-\int _{-\infty }^{\infty }{\frac {dH(x)}{dx}}f(x)\mathrm {d} x=-\int _{-\infty }^{\infty }\delta (x)f(x)\mathrm {d} x=-f(0).$

Here the "turnover rule" for d/dx is used, which may be proved by integration by parts and which holds when f(x) vanishes at the integration limits.