Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $\tau_X$ and $\tau_Y$ be the metric topologies induced by $d_X$ and $d_Y$, respectively. Let $f: X \to Y$ be a map, and let $x_0 \in X$. Then $f$ is continuous at $x_0$ with respect to the topologies $\tau_X$ and $\tau_Y$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that, for every $x \in X$,
\begin{align*}
d_X(x,x_0)<\delta \implies d_Y(f(x),f(x_0))<\varepsilon.
\end{align*}