\begin{definition}[Definition:Heaviside Step Function]
Let $c \ge 0$ be a constant real number.
The '''Heaviside step function on $c$''' is the real function $u_c: \R \to \R$ defined as:
:$\map {u_c} t := \begin{cases}
1 & : t > c \\
0 & : t < c
\end{cases}$
If $c = 0$, the subscript is often omitted:
:$\map u t := \begin{cases}
1 & : t > 0 \\
0 & : t < 0
\end{cases}$
There is no universal convention for the value of $\map {u_c} c$.
However, since $u_c$ is piecewise continuous, the value of $u_c$ at $c$ is usually irrelevant.
\end{definition}