The Heaviside step function is a discontinuous function whose value is $0$ for all negative values and $1$ for non-negative values. It is defined as
$$H(x) \equiv \begin{cases}0 & \text{for } x<0 \ 1 & \text{for } x > 0\end{cases}.$$
For analytical applications, $H$ is usually not defined at $x = 0$, although for many numerical use cases, the average is often used, namely $H(0) = \frac{1}{2}$.