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||. $