\begin{definition}[Definition:Signum Function]
Let $X \subseteq \R$ be a subset of the real numbers.
The '''signum function''' $\sgn: X \to \set {-1, 0, 1}$ is defined as:
:$\forall x \in X: \map \sgn x := \sqbrk {x > 0} - \sqbrk {x < 0}$
where $\sqbrk {x > 0}$ etc. denotes Iverson's convention.
That is:
:$\forall x \in X: \map \sgn x := \begin{cases}
-1 & : x < 0 \\
0 & : x = 0 \\
1 & : x > 0
\end{cases}$
\end{definition}