This problem is in the context of the section on the mean value theorem:Suppose is differentiable on , except possibly at , and is continuous on ; assume exists. Prove that is differentiable at and is continuous at .
The section also talks about Rolle's Theorem and the Cauchy Mean Value Theorem, but they do not appear to be relevant to this problem.If is continuous on and differentiable on , then there is a such that
I've tried attacking this proof from several angles, but I just don't see how the mean value theorem could fit in, or how otherwise to prove it.
Here's what I most recently tried...
Choose and (where will be shortly defined).
Observe that for each and for each , there is or with
Now, we know that
Which is to say that there is such that if then , or
But that just has me going in circles.
Any help would be much appreciated!