I am stuck in following homework question.
Let $f : \mathbb R^n \to \mathbb R^n$ be a uniform contraction and $g(x) = x - f(x)$. Investigate whether $g : \mathbb R^n \to \mathbb R^n$ is a homeomorphism or not.
The definition of uniform contraction is as follows:
$(X,d)$ is metric space. $f: X \to X$ is uniform contraction if there exists $0 <\alpha <1$ such that $d(f(x),f(y)) \leq \alpha d(x,y)$.
I proved that $f$ and $g$ are continuous. But I do not have any idea about inverse of $g$.
By invariance of domain, one could conclude that a bijective continuous function $h: \mathbb R^n \to \mathbb R^n$ is homeomorphism. But I do not know how I can see $g$ is bijective or not. I do not have any other idea to prove homeomorphism. Maybe it is not a homeomorphism but I can't think of any counterexample.
Thank you in advance.