\begin{definition}[Definition:Pointwise Maximum of Mappings/Real-Valued Functions]
Let $S$ be a set.
Let $f, g: S \to \R$ be real-valued functions.
Let $\max$ be the max operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined).
Then the '''pointwise maximum of $f$ and $g$''', denoted $\map \max {f, g}$, is defined by:
:$\map \max {f, g}: S \to \R: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$
'''Pointwise maximum''' thence is an instance of a pointwise operation on real-valued functions.
\end{definition}