Combination Theorem for Continuous Functions/Real/Quotient Rule

Theorem

Let $\R$ denote the real numbers.

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

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 $\map g x \ne 0$.

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:

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

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

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

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 $\map g x \ne 0$.

$\blacksquare$

1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: Exercise $1.5: 19$
1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 9.4 \ \text{(iii)}$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): continuous function (iii)