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directional derivatives and convex sets

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directional derivatives and convex sets

Assume U \subset R^n is convex (not necessarily open) and all partial derivatives (first order) exist in all points of U. Furthermore all partial derivatives are bounded by the same number M >= 0. Prove that for all x, y \in U:
|f(x) - f(y)| <= M \sqrt{k} ||x - y||_2

It is rather easy, when U is multi-dimensional interval (one can estimate |f(x) - f(y)| coordinate-wise), but I can't come up with generalization.

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