Heine–Cantor theorem

Heine–Cantor theorem Not to be confused with Cantor's theorem. This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. (April 2019) (Learn how and when to remove this template message) In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if {displaystyle fcolon Mto N} is a continuous function between two metric spaces {displaystyle M} and {displaystyle N} , and {displaystyle M} is compact, then {displaystyle f} is uniformly continuous. An important special case is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous.

Proof Suppose that {displaystyle M} and {displaystyle N} are two metric spaces with metrics {displaystyle d_{M}} and {displaystyle d_{N}} , respectively. Suppose further that a function {displaystyle f:Mto N} is continuous and {displaystyle M} is compact. We want to show that {displaystyle f} is uniformly continuous, that is, for every positive real number {displaystyle varepsilon >0} there exists a positive real number {displaystyle delta >0} such that for all points {displaystyle x,y} in the function domain {displaystyle M} , {displaystyle d_{M}(x,y)0} . By continuity, for any point {displaystyle x} in the domain {displaystyle M} , there exists some positive real number {displaystyle delta _{x}>0} such that {displaystyle d_{N}(f(x),f(y))

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