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 continous 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 :)