Combination Theorem for Continuous Functions/Real/Quotient 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:
 * $\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$

that is, on all the points $x$ of $S$ where $\map g x \ne 0$.

Proof
By definition of continuous, we have that
 * $\forall c \in S: \displaystyle \lim_{x \mathop \to c} \map f x = \map f c$
 * $\forall c \in S: \displaystyle \lim_{x \mathop \to c} \map g x = \map g c$

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 Quotient Rule for Limits of Functions, we have that:
 * $\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$

wherever $m \ne 0$.

So, by definition of continuous again, we have that $\dfrac f g$ is continuous on all points $x$ of $S$ where $\map g x \ne 0$.