Appendix/Mathematics/StepFunction

Unit Step Function and Its Derivative

Standard Presentation

The unit — or, Heaviside — step function, $H(x)$, is defined such that,

$H(x)=\{\begin{array}{ll}0;& x<0\\ 1;& x>0\end{array}$ [MF53], Part I, §2.1 (p. 123), Eq. (2.1.6)

In evaluating this function at $x=0$, we will adopt the half-maximum convention and set $H(0)=\frac{1}{2}$. As has been pointed out in, for example, a relevant Wikipedia discussion, the derivative of the unit step function is,

$\frac{dH(x)}{dx}$

$=$

$\delta (x),$

where, $\delta (x)$ is the Dirac Delta function. Hence, the unit step function is sometimes written as,

$H(x)$

$=$

$\underset{-\mathrm{\infty }}{\overset{x}{\int }}\delta (\xi )d\xi .$

Sign of a Function

Notice that the sign of $x$, may be written in terms of the step function as,

$\mathrm{sgn}[x]=\frac{|x|}{x}$

$=$

$2H(x)-1.$

Hence,

$\frac{d}{dx}[\mathrm{sgn}(x)]=\frac{d}{dx}\left[\frac{|x|}{x}\right]$	$=$	$2\delta (x).$
📚 Hunter (2003), §2.2, immediately following Eq. (3)

See Also