# Uniform continuity

A property of a function (mapping) , where and are metric spaces. It requires that for any there is a such that for all satisfying , the inequality holds.

If a mapping is continuous on and is a compactum, then is uniformly continuous on . The composite of uniformly-continuous mappings is uniformly continuous.

Uniform continuity of mappings occurs also in the theory of topological groups. For example, a mapping , where , and topological groups, is said to be uniformly continuous if for any neighbourhood of the identity in , there is a neighbourhood of the identity in such that for any satisfying (respectively, ), the inclusion (respectively, ) holds.

The notion of uniform continuity has been generalized to mappings of uniform spaces (cf. Uniform space).

#### References

[1] | A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) |

[2] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |

[3] | J.L. Kelley, "General topology" , Springer (1975) |

[4] | N. Bourbaki, "General topology" , Elements of mathematics , Springer (1989) (Translated from French) |

#### Comments

There are several natural uniform structures on a topological group; the (confusing) statement above about uniform continuity of mappings between them can be interpreted in various ways.

#### References

[a1] | W. Roelcke, S. Dierolf, "Uniform structures on topological groups and their quotients" , McGraw-Hill (1981) |

[a2] | R. Engelking, "General topology" , Heldermann (1989) |

**How to Cite This Entry:**

Uniform continuity.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Uniform_continuity&oldid=12797