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