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I'm reading materials about Mather theory, weak KAM theory. Well, the question is simple;

Assume that $$|u(x+h)-u(x)-Du(x)\cdot h|\leq C|h|^2$$ for all $x\in\mathcal{M}_0,\ h\in\mathbb{R}^n$. The function $u$ is defined on the torus $\mathbb{T}^n$, is known to be only Lipschitz continuous and to be differentiable at all points in $\mathcal{M}_0$. I think knowing $\mathcal{M}_0$ being just a subset of the torus is enough here.

Then how can we know that $$|Du(x)-Du(y)|\leq C|x-y|$$ for all $x,y\in\mathcal{M}_0$?

The material suggests me writing

$$|u(y)-u(x)-Du(x)\cdot(y-x)|\leq C|y-x|^2$$ and $$|u(x)-u(y)-Du(y)\cdot(x-y)|\leq C|x-y|^2,$$ and then combining them with the triangle inequality. I'm stuck here because I only get $$|(Du(x)-Du(y))\cdot(x-y)|\leq C|x-y|^2.$$ I'm not sure it's enough to have the desired conclusion. Anyone helps?

Thanks.

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