Basic differentiability question

[Gyroid]

Homework Statement

Let f: R -> R be a continuous function such that f '(x) exists for all x =/= 0 . Say also that the limit of f '(x) as x goes to 0 exists and is equal to L. Must f '(0) exist as well? Prove or disprove.

The Attempt at a Solution

I can't come up with a proof or counterexample. It seems like it must be true but I've learned to not completely trust my intuition when it comes to these things (pathological counterexamples come to mind). Can anyone give me a hint on whether or not this is true or how to go about proving/disproving it?

 

[Dick]

Think about the difference quotient (f(h)-f(0))/h as h->0. If that does not converge to 0 then the there is a sequence h_i->0 such that the difference quotient is bounded away from 0. Now think about the Mean Value Theorem. Remember that the premises of the MVT don't require differentiability at the endpoints.

 

[Gyroid]

Got it. thanks a lot, that was perfect.