# uniform continuity of Lipschitz functions

###### Proposition 1.

An Hölder continuous mapping is uniformly continuous^{}.
In particular any Lipschitz continuous mapping is uniformly continuous.

###### Proof.

Let $f:X\to Y$ be a mapping such that for some $C>0$ and $\alpha $ with $$ one has

$${d}_{Y}(f(p),f(q))\le C{d}_{X}{(p,q)}^{\alpha}.$$ |

For every given $\u03f5>0$, choose $\delta ={\left(\u03f5/(C+1)\right)}^{\frac{1}{\alpha}}$. If $p,q\in X$ are given points satisfying

$$ |

then

$$ |

as desired. ∎

Title | uniform continuity of Lipschitz functions |
---|---|

Canonical name | UniformContinuityOfLipschitzFunctions |

Date of creation | 2013-03-22 15:06:16 |

Last modified on | 2013-03-22 15:06:16 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 8 |

Author | paolini (1187) |

Entry type | Theorem |

Classification | msc 26A16 |