This is a curious problem I found in the "challenge" section of the text I'm learning real analysis from.
Suppose $A$ is a compact subset of $\mathbb{R}^{n}$ and that $f$ is a continuous function mapping $A$ one-to-one onto $B \subseteq \mathbb{R}^{m}$. Try and prove that the inverse function $f^{-1}$ is continuous on $B$.
I figure that this most likely has to do with the theorem stating that if a subset is compact in a particular vector space, then for a continuous $f$ mapping onto another vector space the function of the subset is compact in this other vector space i.e. the continuous image of a compact set is compact. I tried investigating the proof of this theorem to see if it would give me any intuition as to how to prove this question but wasn't successful, it feels required to solve the problem though.
Best Answer
There is a famous result that a continuous bijection $f$ from compact set into a Hausdorff space is a homeomorphism, i.e. $f^{-1}$ is continuous (on $\mathscr R(A)$, that is). Since the assumption stated that $A$ is compact and since $\Bbb R^n$ is a Hausdorff space for all $n\in \Bbb N$, the result follows.