Combination Theorem for Continuous Functions/Real/Product Rule

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Let $\R$ denote the real numbers.

Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.


$f g$ is continuous on $S$

where $f g$ denotes the pointwise product of $f$ and $g$.


By definition of continuous:

$\forall c \in S: \ds \lim_{x \mathop \to c} \map f x = \map f c$
$\forall c \in S: \ds \lim_{x \mathop \to c} \map g x = \map g c$

Let $f$ and $g$ tend to the following limits:

$\ds \lim_{x \mathop \to c} \map f x = l$
$\ds \lim_{x \mathop \to c} \map g x = m$

From the Product Rule for Limits of Real Functions:

$\ds \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$

So, by definition of continuous again, we have that $f g$ is continuous on $S$.