Heaviside step distribution

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

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

Differentiating the Heaviside distribution

At first glance things are hopeless:

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

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

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

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

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

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

in the limit this is the Dirac delta function. Thus

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

The delta function has the fundamental property that

$\int _{-\infty }^{\infty }f(x)\delta (x-a){\rm {d}}x=f(a)$

References

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