Composition of continuous functions is continuous

Statement

Let $D, E \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$ be a function with $f(D) \subseteq E$, and let $g \colon E \to \mathbb{R}$. If $f$ is continuous at $a \in D$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$. In particular, if $f$ and $g$ are continuous, then $g \circ f$ is continuous.

Remarks

Given $\varepsilon > 0$, choose $\eta > 0$ for $g$ at $f(a)$, and then $\delta > 0$ for $f$ at $a$ and the tolerance $\eta$. See the epsilon-delta form. For example, $x \mapsto \sin(x^2)$ is continuous. Mathlib states it for topological spaces, as Continuous.comp.

Explanations

L1 human

A small change of $x$ gives a small change of $f(x)$. This in turn gives a small change of $g(f(x))$.