Continuity in a Compact Set
Compactness is intimately connected with the properties of continuity, uniform continuity, closedness and boundedness. One such is illustrated in the following theorem.
Letbe continuous withcompact (closed and bounded. Thenis uniformly continuous.
Proof: ChooseSinceis continuous onis continuous atfor eachThus for eachthere issuch that ifandthenConsider the family of intervalsThis is an open cover ofandis compact so there is a finite subcover ofso there aresuch that
Letand supposeandthen there is such thatNowmaking
Henceandis uniformly continuous.
This theorem has a converse. Supposeis uniformly continuous thenhas a limit at each accumulation point ofLetbe the set of accumulation points ofand let defineDefinebyforandfor The functionis continuous.