Combination Theorem for Continuous Real-Valued Functions/Product 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 $f g : S \to \R$ be the pointwise product of $f$ and $g$, that is, $f g$ is the mappping defined by:
- $\forall s \in S : \map {\paren{f g} } s = \map f s \map g s$
Then:
- $f g$ is a continuous real-valued function
Proof
Follows from:
- Real Numbers form Valued Field
- By definition a valued field is a normed division ring
- Product Rule for Continuous Mappings into Normed Division Ring
$\blacksquare$