I'm not sure how to go about proving this theorem:
Let $f:\mathbb{R}^m \longrightarrow \mathbb{R}^m,f\in C^1(\mathbb{R}^m)$ such that:
$\|f'(x)(v)\|=\|v\|,\forall v\in \mathbb{R}^m,\forall x \in \mathbb{R}^m$
Prove that $\|f(x)-f(y)\|=\|x-y\|,\forall x,y\in \mathbb{R}^m$.
Any hints would be appreciated.
By the Theorem of the Mean Value Inequality, we have $\|f(x)-f(y)\|\le\|x-y\|,\forall x,y\in \mathbb{R}^m$.