Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $f:X \to Y$ be a map. Suppose $f$ is uniformly continuous; that is, for every $\varepsilon>0$ there exists $\delta>0$ such that for all $u,v \in X$, if $d_X(u,v)<\delta$, then $d_Y(f(u),f(v))<\varepsilon$. Then $f$ is continuous on $X$; equivalently, for every $a \in X$ and every $\varepsilon>0$, there exists $\eta>0$ such that for every $z \in X$, if $d_X(a,z)<\eta$, then $d_Y(f(a),f(z))<\varepsilon$.