Yes, I forgot the initial conditions. Thanks for pointing that out! I've edited the question.

Do you mind a stupid question? How is the Lemma you provided related to Lemma 3.1.1 in Kleps paper? Kind regards /Olav

In case someone else comes across this. A motivation for why the second condition above is needed is of course given by considering the map $x \rightarrow e^{x}$. Obviously the derivative never vanishes, but $e^{x} = a$ has no solution for $a \leq 0$.

Why do you think they are purely local? The second condition, that the norm of f tends to infinity, does not appear in the usual inverse function theorem

Nonlinear Systems by HK Khalil is a modern classic

Thanks for that! I'll check out the book. It looks as if the Matlab function "symbfact" does precisely the required symbolic factorization in a safe way.