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→Y be a mapping such that for some C>0 and α with 0<α≤1 one has
dY(f(p),f(q))≤CdX(p,q)α. |
For every given ϵ>0, choose δ=(ϵ/(C+1))1α. If p,q∈X are given points satisfying
dX(p,q)<δ |
then
dY(f(p),f(q))≤Cδα≤CϵC+1<ϵ, |
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 |