1617-257/TUT-R-5

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.