Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions

Definition

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.

General Definition

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:

$\max \set {f_1, f_2, \ldots, f_n}: X \to \overline \R : \map {\max \set {f_1, f_2, \ldots, f_n} } x := \begin{cases}\map {f_1} x & n = 1 \\ \max \set {\max \set {\map {f_1} x, \map {f_2} x, \ldots, \map {f_{n - 1}} x }, \map {f_n} x} & n \ge 2\end{cases}$

Also see

• Pointwise Minimum of Extended Real-Valued Functions, an analogous notion tied to the min operation
• Pointwise Operation on Extended Real-Valued Functions for more operations on extended real-valued functions
• Pointwise Maximum of Mappings for the pointwise maximum of more general mappings
• Pointwise Maximum of Real-Valued Functions