Suppose $f :\mathbb{R}^n \rightarrow \mathbb{R}^n $ is of class $C^1$, and $\|f(x)-f(y)\|\geq\|x-y\|$. Prove that $f$ is global invertible, and $f^{-1}$ is also of class $C^1$.
We learnt the Inverse Function Theorem for multi-variable functions, see this . But it only tells us about "local" inverses, not "global" inverses. I found some global type answers such as this answer and this one. But these answers didn't help for my problem. I wonder if any one can provide some clues about my problem?
Let me give you a sketch of a possible argument: First show $f$ is injective. You can do this by definition.
Next show $f$ is surjective. I would recommend showing, that the image of $f$, i.e. $$f(\mathbb{R}^n)\subseteq \mathbb{R}^n,$$ is open, closed and not empty. Since $\mathbb{R}^n$ is connected, we would then have $$f(\mathbb{R}^n)= \mathbb{R}^n,$$ which is surjectivity. To show that the image is open, you can use the 'local' inverse function theorem. The closedness can be done via sequences (Cauchy criterion for convergence) and the assumed inequality on $f$. This shows you, that $f$ is bijective, hence can be inverted.
The regularity of the inverse function follows again from the 'local' inverse function theorem, since differentiability and continuity are local properties.
