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)

As has been pointed out in, for example, the relevant Wikipedia discussion, the derivative of the unit step function is the Dirac Delta function,

and the integral of the delta function is $H$

where, $\delta (\xi )$ is the Kronicher delta function.

See Also