Combination Theorem for Continuous Real-Valued Functions/Maximum Rule
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Theorem
Let $\struct{S, \tau}$ be a topological space.
Let $\R$ denote the real number line.
Let $f, g :S \to \R$ be continuous real-valued functions.
Let $f \vee g$ denote the pointwise maximum of $f$ and $g$, that is, $f \vee g$ is the mapping defined by:
- $\forall s \in S : \map {\paren{f \vee g} } s = \max \set{\map f s, \map g s}$
Then:
- $f \vee g$ is a continuous real-valued function
Proof
From Characterization of Pointwise Maximum of Real-Valued Functions:
- $f \vee g = \dfrac 1 2 \paren{f + g + \size{f - g}}$
We have:
\(\ds \) | \(\) | \(\ds f, g \text{ are continuous real-valued functions}\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds f -g \text{ is a continuous real-valued function}\) | Difference Rule for Continuous Real-Valued Functions | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \size{f - g} \text{ is a continuous real-valued function}\) | Absolute Value Rule for Continuous Real-Valued Function | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds f + g + \size{f - g} \text{ is a continuous real-valued function}\) | Sum Rule for Continuous Real-Valued Functions | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \dfrac 1 2 \paren{f + g + \size{f - g} } \text{ is a continuous real-valued function}\) | Multiple Rule for Continuous Real-Valued Function |
The result follows.
$\blacksquare$