uniformly continuous

Let f:A be a real function defined on a subset A of the real line. We say that f is uniformly continuousPlanetmathPlanetmath if, given an arbitrary small positive ε, there exists a positive δ such that whenever two points in A differ by less than δ, they are mapped by f into points which differ by less than ε. In symbols:


Every uniformly continuous function is also continuousMathworldPlanetmath, while the converse does not always hold. For instance, the functionMathworldPlanetmath f:]0,+[ defined by f(x)=1/x is continuous in its domain, but not uniformly.

A more general definition of uniform continuity applies to functions between metric spaces (there are even more general environments for uniformly continuous functions, i.e. uniform spaces). Given a function f:XY, where X and Y are metric spaces with distances dX and dY, we say that f is uniformly continuous if


Uniformly continuous functions have the property that they map Cauchy sequencesPlanetmathPlanetmath to Cauchy sequences and that they preserve uniform convergenceMathworldPlanetmath of sequences of functions.

Any continuous function defined on a compact space is uniformly continuous (see Heine-Cantor theorem).

Title uniformly continuous
Canonical name UniformlyContinuous
Date of creation 2013-03-22 12:45:38
Last modified on 2013-03-22 12:45:38
Owner n3o (216)
Last modified by n3o (216)
Numerical id 14
Author n3o (216)
Entry type Definition
Classification msc 26A15
Related topic UniformContinuity
Defines uniformly continuous function