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On 10/13/16, we proved that if U is an open and convex subset of \mathbb R^n and if f : U \to \mathbb R is differentiable with \|D f (x)\| \leq M for all x \in U then we have that |f(x) - f(y)| \leq M \|x - y\| for all x, y \in U.

We also proved the analogous statement if f is Lipschitz continuous instead of having uniformly bounded derivative.

Lastly, we created a formulation for the problem if U is star-shaped rather than convex.