Recently, I learn about the Hadamard global inverse function theorem. It says that if $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is $C^2$ and proper (preimage of compact set is again compact) with non-vanishing Jacobian, then $f$ admits a global inverse.
Now, I have a question that given a smooth function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that $|\det(Df)(x)|\geq\delta>0$ and $f(x)$ has bounded derivative up to all orders, whether $f$ is proper or $f$ admits a global inverse?
I have no any idea to construct a counterexample or prove it. Any opinion is welcome and thanks in advanced.