# uniformly continuous

Let $f:A\to \mathbb{R}$ be a real function defined on a subset $A$ of the real line. We say that $f$ is *uniformly continuous ^{}* if, given an arbitrary small positive $\epsilon $, there exists a positive $\delta $ such that whenever two points in $A$ differ by less than $\delta $, they are mapped by $f$ into points which differ by less than $\epsilon $. In symbols:

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Every uniformly continuous function is also continuous^{}, while the converse does not always hold. For instance, the function^{} $f:]0,+\mathrm{\infty}[\to \mathbb{R}$ 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 $f:X\to Y$, where $X$ and $Y$ are metric spaces with distances ${d}_{X}$ and ${d}_{Y}$, we say that $f$ is uniformly continuous if

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Uniformly continuous functions have the property that they map Cauchy sequences^{} to Cauchy sequences and that they preserve uniform convergence^{} of sequences of functions.

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

Title | uniformly continuous |
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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 |