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Let $U,V \subseteq \mathbb{R^n}$ be open. Suppose that $f: U \rightarrow V$ is surjective and is everywhere Frechet differentiable with $df(x)$ invertible for all $x \in U$. Is $f$ invertible?

I know that one of the conditions for the inverse function theorem is that $f$ is continuously differentiable, which we don't necessarily have. If the question didn't give us that $f$ was surjective, I'd have concluded that $f$ doesn't have to be invertible, but I'm just wondering what the question is trying to tell us by mentioning surjectivity of $f$, if it implies that $f$ is continuously differentiable I don't see how.

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No, and the issue is not the continuity of the derivative. To see this, you could take the function from wikipedia's article on the inverse function theorem:

$$F(x)=\left( \begin{matrix} e^x \cos(y) \\ e^x \sin(y) \end{matrix} \right),$$ which is a surjective continuously differentiable function from $\mathbf{R}^2$ to $\mathbf{R}^2 \setminus \{0 \}$ whose differential is everywhere invertible. This has an inverse everywhere locally, but no global inverse since it is periodic in $y$.

(This is really the holomorphic function $z \mapsto e^z$ from $\mathbf{C}$ to $\mathbf{C}^\times$ in disguise).

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  • $\begingroup$ ahh yes of course forgot about periodicity, still confused as to why questions mentioned surjective though, does surjectivity imply local inverse? $\endgroup$ – Displayname Feb 14 at 12:26
  • $\begingroup$ If the question wanted to get the reader to appreciate the fact that periodic functions don't have a global inverse, then why not give us that $f$ is continuously differentiable too? $\endgroup$ – Displayname Feb 14 at 12:37
  • $\begingroup$ won't let me at, @Stephen $\endgroup$ – Displayname Feb 14 at 12:38
  • $\begingroup$ @Displayname I would be very surprised if surjective plus differentiable implies continuously differentiable. I don't know why the question was phrased the way it was! You don't need to @ me when you reply to one of my posts: I automatically get notified. $\endgroup$ – Stephen Feb 14 at 16:28

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