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 onehas

$$d_Y(f(p),f(q))\le C{d}_X{(p,q)}^{\alpha }.$$

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

then

as desired.∎