I am unsure of this proof, and I would appreciate any input about whether my reasoning is on the right track. Due tomorrow btw.

The question is:

Suppose that is continuous everywhere, exists for , and exists. Prove that exists.

Professor said to use L'Hôpital's rule if we get stuck.

I started by trying to let and working through an argument using L'Hôpital's rule, but I made some extra assumptions and such which dont seem to be reasonable. Any nudges in the right direction would be great. My work is pretty useless...and ultimately it seems like I wouldnt be proving the right thing anyway. I feel like I am overthinking (as usual) what would be an otherwise simple and elegant proof. If anyone really wants to see my attempt, I will post it, but it is almost certainly incorrect. Thanks in advance.

edit: I know intuitively that it must exist, since is continuous and exists everywhere but (for sure). And since the two-sided limit exists at , either is continuous at or has a removable discontinuity there, but it must be defined there because is continuous at and does not have a corner there. There cannot be a corner at because the two-sided limit exists, so we know that the derivative must exist there (as is continuous without a corner)... but I do not see how L'Hôpital would come into play in a formal proof?