Definition:Pointwise Maximum of Mappings/Real-Valued Functions

Definition
Let $S$ be a set.

Let $f, g: S \to \R$ be real-valued functions.

Let $\max$ be the max operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined).

Then the pointwise maximum of $f$ and $g$, denoted $\map \max {f, g}$, is defined by:


 * $\map \max {f, g}: S \to \R: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$

Pointwise maximum thence is an instance of a pointwise operation on real-valued functions.

Also see

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