In another question, I ask about the topology of the singular manifold of the Jacobian. What i want to ask in here is about the radius of convergence of a Taylor series expansion of the inverse function. In particular;
is the domain of covergence always a ball? (GEdgar answer points that this is not so in general with a very simple counterexample)
what determines the shape of the domain of convergence in general? is it only bounded by the singular manifold of the Jacobian?
Edit I did an inexcusable mistake in this question; i completely forgot to mention the very important detail that what i am asking here is for what valid extensions exists of the inverse function theorem; in this regard i wanted to gather if the domain of convergence is going to be only bounded by the singular jacobian manifold, or if in general it can be smaller (so the original theorem is all it can be said with generality)