I am studying the following proof for which an excerpt is provided below:
Update: I have written out a fully-detailed proof of an argument that seeks verify the claim that $\partial \psi$ is invertible. (1) I am unclear on what is special about the point $(x_0, 0)$ as the proof seems to goes through irrespective of the value of the particular point and (2) The author's logic at the end seems reversed to me. It would be helpful if someone could critique the proof below and indicate what step, if any, is incorrect. Also, I apologize for not including the actual TEX; it was formulated locally and I made use of many macros that mathjax wouldn't understand: