\begin{definition}[Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions/General Definition]
Let $X$ be a set.
Let $\max$ be the max operation on $\overline \R$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined).
Let $f_1, f_2, \ldots, f_n : X \to \overline \R$ be extended real-valued functions.
Then the '''pointwise maximum of $f_1, f_2, \ldots, f_n$''', denoted $\max \set {f_1, f_2, \ldots, f_n}$, is defined by:
:$\max \set {f_1, f_2, \ldots, f_n}: X \to \overline \R : \map {\max \set {f_1, f_2, \ldots, f_n} } x := \begin{cases}\map {f_1} x & n = 1 \\ \max \set {\max \set {\map {f_1} x, \map {f_2} x, \ldots, \map {f_{n - 1}} x }, \map {f_n} x} & n \ge 2\end{cases}$
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\end{definition}