Let me ask a question which appears in the book 'Elementary Differential Geometry' written by O'Neil.
The questions is: prove that if a one-to-one and onto mapping $f:\Bbb R ^n \to \Bbb R ^n$ is regular, then it is diffeomorphism.
In the book, "$f$ is regular" means that the tangent map of $f$ is one to one. I think certainly I need to use the inverse function thorem. $f$ is a mapping so $f$ is in the class $\mathcal C ^1$. Since $f$ is regular and in the class $\mathcal C^1$, its Jacobian is invertible, so the derivative of $f$ is invertible.
Therefore, I might apply the inverse function theorem to $f$: there exists an open set in $\Bbb R ^n$ in which the inverse of $f$ exists and the inverse belongs to the class $\mathcal C ^1$ so $f$ is a diffeomorphism. But to prove $f$ is a diffeomorphism, shouldn't I show the inverse of $f$ is in the class $\mathcal C ^1$ for every point in $\Bbb R ^n$?
I would really appreciate any help. Thank you for reading.