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