\begin{definition}[Definition:Pointwise Inequality of Extended Real-Valued Functions]
Let $S$ be a set, and let $f, g: S \to \overline \R$ be extended real-valued functions.
Then '''pointwise inequality of $f$ and $g$''', denoted $f \le g$, is defined to hold {{iff}}:
:$\forall s \in S: \map f s \le \map g s$
where $\le$ denotes the usual ordering on the extended real numbers $\overline \R$.
Thence '''pointwise inequality''' of extended real-valued functions is an instance of an induced relation on mappings.
\end{definition}