Appendix/Mathematics/StepFunction

Unit Step Function and Its Derivative

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,

$H(x)$

$=$

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

See Also