Uniform Continuity
In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between x and y cannot depend on x and y themselves. For instance, any isometry (distance-preserving map) between metric spaces is uniformly continuous.
Every uniformly continuous function between metric spaces is continuous. Uniform continuity, unlike continuity, relies on the ability to compare the sizes of neighbourhoods of distinct points of a given space. In an arbitrary topological space this may not be possible. Instead, uniform continuity can be defined on a metric space where such comparisons are possible, or more generally on a uniform space.
The equicontinuity of a set of functions is a generalization of the concept of uniform continuity.
Every continuous function on a compact set is uniformly continuous.
Read more about Uniform Continuity: Definition For Functions On Metric Spaces, Local Continuity Versus Global Uniform Continuity, Examples, Properties, History, Relations With The Extension Problem, Generalization To Topological Vector Spaces, Generalization To Uniform Spaces
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