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 ).