Example of function $f $, that is differentiable at $0$, and has inverse function $g$ , that has jump discontinuity at $g(f(0))$? So first of all I am sorry if this is breaking any posting rules. Yesterday i asked a question  Example of function that is differentiable at $0$, and has inverse function that is not continuous at $0$? , and since people asked for more explanation and it seems i cant edit the question i decided to post a new one.
So the first thing i stated wrong in my question is i need inverse function that is not continuous at $f(0)\neq 0$... And also that discontinuity has to be jump discontinuity.
I would also like to thank everyone who helped me when i first asked the question, and since this is getting longer than it should i will sum it up below by repeating the title of the question:
I need to find example of function $f$, that is Differentiable at $0$, and has inverse function $g$, that has  jump discontinuity at $g(f(0))$.
Thanks for reading :)

This is not possible, as stated.  For $g$ to be the inverse function of $f$ on some neighborhood of $x=0$, by definition $g(f(x)) = f(g(x)) = x$ on that neighborhood.  The map $x \mapsto x$ does not have a jump discontinuity.
