1617-257/TUT-R-5

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

We also proved the analogous statement if [math]\displaystyle{ f }[/math] is Lipschitz continuous instead of having uniformly bounded derivative.

Lastly, we created a formulation for the problem if [math]\displaystyle{ U }[/math] is star-shaped rather than convex.