Combination Theorem for Continuous Functions/Real/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 $S$.