Continuity is equivalent to sequential continuity in metric spaces

Statement

Let $X$ and $Y$ be metric spaces and let $f \colon X \to Y$ be a function. Then $f$ is continuous if and only if $f(x_n) \to f(x)$ for each sequence $(x_n)$ in $X$ that converges to a point $x$.

Remarks

The condition on sequences is sequential continuity. Proof of the converse: if $f$ is not continuous at $x$, then by epsilon-delta continuity there are $\varepsilon > 0$ and points $x_n$ with $d(x_n, x) < 1/n$ and $d(f(x_n), f(x)) \ge \varepsilon$. Then $x_n \to x$, but $f(x_n) \not\to f(x)$. The same equivalence holds at each single point. In general topological spaces, continuity implies sequential continuity, but the converse can fail.

Explanations

L1 human

A continuous map sends convergent sequences to convergent sequences, with the right limit. To show that a map is not continuous, it is often enough to find one sequence $x_n \to x$ with $f(x_n) \not\to f(x)$.