uniform continuity of Lipschitz functions

An Hölder continuous mapping is uniformly continuous. In particular any Lipschitz continuous mapping is uniformly continuous.

Let $f\colon X\to Y$ be a mapping such that for some $C>0$ and $\alpha$ with $0<\alpha\leq 1$ one has

d_{Y}(f(p),f(q))\leq Cd_{X}(p,q)^{\alpha}.

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

d_{X}(p,q)<\delta

then

d_{Y}(f(p),f(q))\leq C\delta^{\alpha}\leq C\frac{\epsilon}{C+1}<\epsilon,

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