Uniform Continuity - Definition For Functions On Metric Spaces

Definition For Functions On Metric Spaces

Given metric spaces (X, d1) and (Y, d2), a function f : XY is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, yX with d1(x, y) < δ, we have that d2(f(x), f(y)) < ε.

If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, yX, |xy| < δ implies |f(x) − f(y)| < ε.

The difference between being uniformly continuous, and simply being continuous at every point, is that in uniform continuity the value of δ depends only on ε and not on the point in the domain.

Read more about this topic:  Uniform Continuity

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