# Heaviside step distribution

The Heaviside step distribution is defined by (Abramowitz and Stegun Eq. 29.1.3, p. 1020):

${\displaystyle H(x)=\left\{{\begin{array}{ll}0&x<0\\{\frac {1}{2}}&x=0\\1&x>0\end{array}}\right.}$

Note that other definitions exist at ${\displaystyle H(0)}$, for example ${\displaystyle H(0)=1}$. In the famous Mathematica computer package ${\displaystyle H(0)}$ is unevaluated.

## Differentiating the Heaviside distribution

At first glance things are hopeless:

${\displaystyle {\frac {{\rm {d}}H(x)}{{\rm {d}}x}}=0,~x\neq 0}$
${\displaystyle {\frac {{\rm {d}}H(x)}{{\rm {d}}x}}=\infty ,~x=0}$

however, lets define a less brutal jump in the form of a linear slope such that

${\displaystyle H_{\epsilon }(x-a)={\frac {1}{\epsilon }}\left(R(x-(a-{\frac {\epsilon }{2}}))-R(x-(a+{\frac {\epsilon }{2}}))\right)}$

in the limit ${\displaystyle \epsilon \rightarrow 0}$ this becomes the Heaviside function ${\displaystyle H(x-a)}$. However, lets differentiate first:

${\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}H_{\epsilon }(x-a)={\frac {1}{\epsilon }}\left(H(x-(a-{\frac {\epsilon }{2}}))-H(x-(a+{\frac {\epsilon }{2}}))\right)}$

in the limit this is the Dirac delta distribution. Thus

${\displaystyle {\frac {\rm {d}}{{\rm {d}}x}}[H(x)]=\delta (x)}$.

## References

1. Milton Abramowitz and Irene A. Stegun "Handbook of Mathematical Functions" Dover Publications ninth printing.