Let and be metric spaces, , , and . We say is continuous at if for every , there exists a such that for all which . If is continuous at every point of , then is continuous on .
Transitivity
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Let be metric spaces, , and , . Define the function . If is continuous at and is continuous at , then is continuous at .
Let . By definition, there is such that for all , implies . Similarly, there is such that for all , implies . Thus implies , and is continuous at .
Topological Characterization of Continuity
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Let be metric spaces. is continuous if and only if is open in for every open set .
Suppose is continuous on and is open. Suppose and . There is some where . By continuity, there is some where . Thus for every , , and is open.
Suppose is open in for every open . For every and , is an open set, so is open by assumption. But is contained in this set, so there is some where . But implies , thus is continuous.
Continuity and Compactness
Preservation of Compactness
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If is a continuous mapping of a compact metric space to metric space , then is compact.
Let be an open cover of .
By Topological Characterization of Continuity, is an open cover of . Since was originally compact, there is a finite subcover . Then and is compact.
Uniform Continuity
Let be a mapping of metric spaces. We say is uniformly continuous on if for every , there exists such that every satisfies implies .