Q:

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment:

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m $ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

A hint my tutor gave me is to use the MVT, which is why I'm trying to get an expression I can use it on. Since $ K $is compact and $f$ is continuously differentiable, the derivative will be bounded since we're in $\mathbb{R}$. I have several problems, firstly I still have no clue how/where to use the MVT and don't know where the fact that K is convex becomes important. (I do know the definition of convex, though.)
I'd appreciate any help.

A:

Hint
Consider two points $x,y\in K.$ Since $K$ is convex you have a segment joining $x$ and $y$ which is contained in $K.$ Now apply the mean value theorem and use that derivatives are bounded on the compact set $K.$

