absolute value function

The sign function assigns to each real number its sign, disregarding its magnitude. The function is defined as follows:

\[
\operatorname{sgn}(x) =
\begin{cases}
-1 & \text{if } x < 0 \\[6pt] 0 & \text{if } x = 0 \\[6pt] 1 & \text{if } x > 0
\end{cases}
\quad \forall \\, x \in \mathbb{R}
\]