Uniform continuity implies continuity

Uniform continuity gives a single delta that works at every point, hence pointwise continuity

Uniform continuity implies continuity

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and let $f:X\to Y$ .

Proposition: If $f$ is uniformly continuous on $X$ , then $f$ is continuous at every point of $X$ .

Uniform continuity is strictly stronger than continuity in general, but on compact sets they coincide (Heine–Cantor ).

Proof sketch: Fix $x_0\in X$ and $\varepsilon>0$ . Uniform continuity gives $\delta>0$ such that $d_X(x,y)<\delta$ implies $d_Y(f(x),f(y))<\varepsilon$ for all $x,y\in X$ . Apply this with $y=x_0$ to get continuity at $x_0$ .