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:

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon \rightarrow 0} this becomes the Heaviside function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(x-a)} . However, lets differentiate first:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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 function. Thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{{\rm d}}{{\rm d}x} [H(x)]= \delta(x)}

The delta function has the fundamental property that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.