# Combination Theorem for Continuous Functions/Product Rule

## 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:

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

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

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

From the Product Rule for Limits of Functions:

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

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

$\blacksquare$