Combination Theorem for Continuous Mappings/Metric Space/Product Rule
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Let $\R$ denote the real numbers.
Let $f: M \to \R$ and $g: M \to \R$ be real-valued functions from $M$ to $\R$ which are continuous on $M$.
Then:
- $f g$ is continuous on $M$.
Proof
By definition of continuous:
- $\forall a \in M: \ds \lim_{x \mathop \to a} \map f x = \map f a$
- $\forall a \in M: \ds \lim_{x \mathop \to a} \map g x = \map g a$
Let $f$ and $g$ tend to the following limits:
- $\ds \lim_{x \mathop \to a} \map f x = l$
- $\ds \lim_{x \mathop \to a} \map g x = m$
From the Product Rule for Limits of Real Functions, we have that:
- $\ds \lim_{x \mathop \to a} \paren {\map f x \map g x} = l m$
So, by definition of continuous again, we have that $f g$ is continuous on $M$.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 14$