Exercise 513Topology › Metric and normed spacesL1Added 3 Nov 2026

Continuity in metric spaces

#continuity#epsilon-delta#metric

Statement

Define continuity of $f:X\to Y$ between metric spaces.

Hint

For all $x_0$ , $\forall\varepsilon>0,\exists\delta>0:\ d_X(x,x_0)<\delta\Rightarrow d_Y(f(x),f(x_0))<\varepsilon$ .

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Definition. $f$ is continuous at $x_0$ iff $\forall\varepsilon>0,\exists\delta>0:\ d_X(x,x_0)<\delta\Rightarrow d_Y(f(x),f(x_0))<\varepsilon$ .

Global. $f$ is continuous iff continuous at every point.

Equivalence. Equivalent to: $f^{-1}(U)$ open in $X$ for every open $U\subseteq Y$ ; and to sequential continuity.

Conclusion. Continuity has multiple equivalent characterizations in metric spaces.