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Let $U$ be a convex open set in $\mathbb{R}^n$ and $f:U\longrightarrow \mathbb{R}$ such that there is $M>0$ with the following property:

$\forall x\in U , \exists r >0: \forall a,b \in B(x,r),|f(a)-f(b)|\le M\cdot ||a-b||$ then it holds that

$\forall x,y\in U,|f(x)-f(y)|\le M\cdot ||x-y||. $

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    As I already commented on a previous question of yours: **To get the best possible answers, you should explain what your thoughts on the problem are so far**. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. Also, many would consider your post rude because it is a command ("Prove..."), not a request for help, so please consider rewriting it.2012-07-08

2 Answers 2