The difference between being uniformly continuous, and being simply continuous at every point, is that in uniform continuity the value of " ? " depends only on " ? " and not on the point in the domain.
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More precisely, this theorem states that the uniform limit of " uniformly continuous " functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.
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An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
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In the special case of two topological vector spaces V and W, the notion of uniform continuity of a map f : V \ to W becomes : for any neighborhood B of zero in W, there exists a neighborhood A of zero in V such that v _ 1-v _ 2 \ in A implies f ( v _ 1 )-f ( v _ 2 ) \ in B.
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In fact this will hold if and only if " f " is uniformly continuous on " D " : for this is true if it has a continuous extension to the closure of " D "; and, if " f " is uniformly continuous, it is easy to check " f " has limits on the unit circle and the same inequalities for uniform continuity hold on the closure of " D ".
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