\begin{definition}[Definition:Pointwise Supremum of Extended Real-Valued Functions]
Let $S$ be a set.
Let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$ be an $I$-indexed collection of extended real-valued functions.
Then the '''pointwise supremum of $\family {f_i}_{i \mathop \in I}$''', denoted $\ds \sup_{i \mathop \in I} f_i: S \to \overline \R$, is defined by:
:$\ds \map {\paren {\sup_{i \mathop \in I} f_i} } s := \sup_{i \mathop \in I} \map {f_i} s$
where the latter supremum is taken in the extended real numbers $\overline \R$.
By Extended Real Numbers form Complete Poset, this supremum is guaranteed to exist.
Thence it can be seen that '''pointwise supremum''' is an instance of a pointwise operation on extended real-valued functions.
\end{definition}