Combination Theorem for Continuous Functions/Product Rule

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Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

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


Then:

$f g$ is continuous on $S$

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


Proof

By definition of continuous:

$\forall c \in S: \displaystyle \lim_{x \mathop \to c} \ f \left({x}\right) = f \left({c}\right)$
$\forall c \in S: \displaystyle \lim_{x \mathop \to c} \ g \left({x}\right) = g \left({c}\right)$


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

$\displaystyle \lim_{x \mathop \to c} \ f \left({x}\right) = l$
$\displaystyle \lim_{x \mathop \to c} \ g \left({x}\right) = m$


From the Product Rule for Limits of Functions:

$\displaystyle \lim_{x \mathop \to c} \ \left({f \left({x}\right) g \left({x}\right)}\right) = l m$


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

$\blacksquare$

Sources