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

$\blacksquare$