Heaviside step Function

Defined as $H(t)=\{\begin{array}{ll}1& \text{for }t>0\\ 0& \text{for }t<0\end{array}$, usually where $H(0)=\frac{1}{2}$, too; originally defined to help model within electrical engineering.

The step function and the $\delta$ function are related by $H^{\prime }(t)=\delta (t)$, since the gradient at the middle of the step function is infinite at $t$.