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.
22.
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.
23.
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.
24.
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.
25.
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 ".
