"Counterexample" for the Inverse function theorem
Solution 1:
Actually, this is not possible in $\mathbb{R}^n$ either.
Indeed, if you have any $\mathscr{C}^1$ injective function $f: \Omega \rightarrow \mathbb{R}^n$, then $f$ is open and a homeomorphism on its image (invariance of domain : https://en.m.wikipedia.org/wiki/Invariance_of_domain ).
From Sard’s theorem (https://en.m.wikipedia.org/wiki/Sard%27s_theorem ), the set of critical values has null measure in $\mathbb{R}^n$, thus has empty interior, thus the set of critical points has no interior as well.