Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be bi-Lipschitz, that is, there is $C>0$ such that $$ C^{-1} |x-y| \le |f(x)-f(y)| \le C |x-y| $$ for all $x,y\in\mathbb{R}^d$.
Is $f$ automatically surjective (and therefore a homeomorphism of $\mathbb{R}^d$)? This is easily seen to be true for $d=1$, and appears to be true in any dimension, but I can't find a proof.
Bi-Lipschitz maps preserve Cauchy sequences and converging sequences, both ways. Hence, they preserve completeness. It follows that $f(\mathbb{R}^d)$ is complete in the restriction metric, hence is a closed subset of $\mathbb{R}^d$.
On the other hand, $f(\mathbb{R}^d)$ is open by the invariance of domain.