Heaviside step distribution

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

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

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


Differentiating the Heaviside distribution[edit]

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 distribution. Thus

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


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