Maximum Rule for Continuous Functions
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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
$\blacksquare$