2

I'm trying to solve this problem: Let $f\colon V \to W$ be a continuous and bijective function, where $V$ and $W$ are open subsets of $\mathbb{R}^n$. Prove that $g=f^{-1}$ is continuous.

I saw that there is a theorem called Invariance of domain and with this I can conclude that $f$ is open and therefore an homeomorphism (and with this $g$ is continuous). But in our course we haven't talked about this theorem. Also I proved that, if for example, $V$ is bounded then also $g=f^{-1}$ is continuous, since in this case $f$ will be closed using the fact that every closed subset of $V$ is compact. But in the general case I don't know how to prove this fact. Is there another option different from Invariance of domain theorem? Could you please give me some suggestion? Thanks.

Inb0_Q
  • 41
  • 4
  • Invariance of domain only assumes that $f$ is a continuous injection. – J. De Ro Jun 26 '20 at 22:49
  • Yes, but since $f$ is a bijection in particular is injective. I don't know if there is another way to prove this. Could you recommend me something? – Inb0_Q Jun 26 '20 at 22:53
  • 3
    I can't. I believe there is no elementary proof. – J. De Ro Jun 26 '20 at 23:01
  • So the unique way is with Invariance of domain? – Inb0_Q Jun 27 '20 at 02:39
  • Unique way: No. But a proof will definitely use advanced techniques like algebraic topology. There may be other proof techniques I'm not aware of. See also here: https://math.stackexchange.com/questions/59532/bijective-continuous-function-on-mathbb-rn-not-homeomorphism?noredirect=1&lq=1 – J. De Ro Jun 27 '20 at 08:33

0 Answers0