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On compact domains, continuity automatically upgrades to uniform continuity
Corollary (Heine–Cantor): Let be a compact metric space and let be a metric space. If is continuous, then is uniformly continuous on .
Connection to parent theorem: This is precisely the Heine–Cantor theorem.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
Let be a function between metric spaces, and let . We say is continuous on if is continuous at every point (equivalently, the restriction is continuous).
Spelled out: for every and every , there exists such that for all ,
Let and . The function is uniformly continuous on if
Compared to ordinary continuity on , the key point is that depends only on , not on the location in .
Useful properties:
Examples in :
The Heine-Cantor theorem states: if is a continuous function from a compact metric space to a metric space , then is uniformly continuous.
For every , there exists such that for all :
The key point is that depends only on , not on the choice of points .