Maximum Rule for Continuous Functions

Theorem
Let $\struct {S, \tau}$ be a topological space.

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

Let $\max \set {f, g}: S \to \R$ denote the pointwise maximum of $f$ and $g$.

Then:
 * $\max \set {f, g}$ is continuous.

Proof
From Sum Less Minimum is Maximum:
 * $\forall x \in S : \max \set {\map f x, \map g x} = \map f x + \map g x - \min \set {\map f x, \map g x}$

Thus:
 * $\max \set {f, g} = f + g - \min \set{f, g}$

From Minimum Rule for Continuous Functions:
 * $\min \set {f, g}$ is continuous

From Multiple Rule for Continuous Mappings into Topological Ring:
 * $-\min \set {f, g}$ is continuous

From Sum Rule for Continuous Mappings into Topological Ring:
 * $f + g -\min \set {f, g}$ is continuous

Thus:
 * $\max \set {f, g}$ is continuous

Also see

 * Minimum Rule for Continuous Functions