I was asked at homework to prove that a continuously differentiable function from $\mathbb{R}^n$ to $\mathbb{R}^m$ on a compact set (the product space of $[-1,1]$ $n$ times to be exact) is Lipschitz. I know how to prove that for $1$ dimension but I don't understand how to formally use the Mean Value Theorem in higher dimensions (for partial derivatives if needed).
Any help would be appreciated.
If $a$, $b\in[-1,1]^n$ then the segment $$\sigma:\quad t\mapsto (1-t)a+tb\qquad(0\leq t\leq1)$$ connecting $a$ with $b$ is contained in this cube as well. It follows that the auxiliary function $$\phi(t):=f\bigl((1-t)a+tb\bigr)$$ is welll defined and $C^1$ on $[0,1]$. We therefore have $$f(b)-f(a)=\phi(1)-\phi(0)=\int_0^1\phi'(t)\>dt\ .$$ Since by the chain rule $\phi'(t)=df\bigl((1-t)a+tb\bigr).(b-a)$ we obtain $$|f(b)-f(a)|\leq\int_0^1\|df\bigl((1-t)a+tb\bigr)\|\>|b-a|\>dt\leq M\,|b-a|\ ,$$ whereby $$M:=\max_{0\leq t\leq 1}\|df\bigl((1-t)a+tb\bigr)\|\leq\max_{x\in[-1,1]^n}\|df(x)\|\ .$$