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