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 $\|Df(x)\|\leq M$ 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.