Combination Theorem for Bounded 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 bounded 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 bounded continuous real-valued function
Proof
Follows from:
$\blacksquare$