The classic Inverse Function Theorem is all about local invertibility of the $C^1$ function at point $p$ with non zero Jacobian determinant in a neighborhood of that point. How we should strengthen assumptions to have:

  • global invertibility of a given function
  • global existence of its derivative
  • continuity of that derivative

There is a global version due to Hadamard (for a reference see Schwartz's Nonlinear Functional Analysis). One formulation is as follows. Suppose $f: \mathbf{R}^n \to \mathbf{R}^n$ is smooth with bounded partial derivatives and $|\det Df| \ge \delta > 0$. Then $f$ is a global diffeomorphism of $\mathbf{R}^n$.

The basic idea is that $f$ maps balls to approximate balls of comparable size (the first condition gives an upper bound for how much $f$ can expand regions, while the the second one limits how much $f$ can squash along any direction), from which you can deduce that $f$ is actually a covering map. Hence $f$ is a diffeomorphism since $\mathbf{R}^n$ is simply connected.


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