# 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 $f\colon X\to Y$ 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 UniformContinuityOfLipschitzFunctions 2013-03-22 15:06:16 2013-03-22 15:06:16 paolini (1187) paolini (1187) 8 paolini (1187) Theorem msc 26A16