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.

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