Composite Continuous Function

Question

Let $g$ and $h$ be real-valued functions with domains $\operatorname{dom}(g)$ and $\operatorname{dom}(h)$ respectively. Suppose that $g$ maps $\operatorname{dom}(g)$ into $\operatorname{dom}(h)$, that $g$ is continuous at $a ∈ \operatorname{dom}(g)$, and that h is continuous at $b = g(a) ∈ \operatorname{dom}(h)$. Show that the composite function $f = h \circ g$, defined by $f(t) = h(g(t)), t ∈ \operatorname{dom}(g)$ is continuous at $a$.

Answer 1

Let $(x_n)$ be a sequence converging to $a$. Then, by the sequence definition of continuous, $$\lim_{n\to\infty}h(g(x_n))=h(\lim_{n\to\infty}g(x_n))=h(g(\lim_{n\to\infty}x_n))$$ Since $h,g$ are continuous.