Proof that a quotient of continuous functions is continuous

Proof

We use sequential continuity and the quotient rule for limits of sequences.

1. Let $D \subseteq \R$ and let $f, g \colon D \to \R$ be continuous with $g(x) \ne 0$ for all $x \in D$. Then $f/g$ is defined on all of $D$.
2. By the equivalence of continuity and sequential continuity, $f$ and $g$ are sequentially continuous.
3. Let $(x_n)$ be a sequence in $D$ that converges to a point $a \in D$. By step 2, $f(x_n) \to f(a)$ and $g(x_n) \to g(a)$.
4. We have $g(a) \ne 0$. By the quotient rule for limits, there is $N$ such that $(f(x_n)/g(x_n))_{n \ge N}$ converges to $f(a)/g(a)$.
5. Let $\varepsilon > 0$. By the epsilon-N form for the sequence of step 4, there is $N' \ge N$ with $\abs{f(x_n)/g(x_n) - f(a)/g(a)} < \varepsilon$ for all $n \ge N'$.
6. The same $N'$ works for the whole sequence $(f(x_n)/g(x_n))_{n \ge 0}$, which is defined by step 1. So $(f/g)(x_n) \to (f/g)(a)$.
7. Since $(x_n)$ and $a$ were arbitrary, $f/g$ is sequentially continuous. By the equivalence again, $f/g$ is continuous.