Combination Theorem for Continuous Real-Valued Functions/Minimum 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 $f \wedge g$ denote the pointwise minimum of $f$ and $g$, that is, $f \wedge g$ is the mapping defined by:

$\forall s \in S : \map {\paren{f \wedge g} } s = \min \set{\map f s, \map g s}$

Then:

$f \wedge g$ is a continuous real-valued function

Proof

From Characterization of Pointwise Minimum 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

\(\ds \)

\(\leadsto\)

\(\ds g - \size{f - g} \text{ is a continuous real-valued function}\)

Difference Rule for Continuous Real-Valued Functions

\(\ds \)

\(\leadsto\)

\(\ds f + g - \size{f - g} \text{ is a continuous real-valued function}\)

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.

$\blacksquare$