I've been stating at this problem for a while and have no idea where to even start. Not looking for an answer, but really just a nudge in the right direction.
The homework is in a section covering Rolle's Theorem and 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 .
Since exists the second part of the question follows immediately. This shows is sequentially continuous and is therefore continuous.
For the first part, since is continuous on and differentiable on then, by MVT, .
We already know that from the question. Therefore we have:
From the question and are defined ( and are defined since is continuous and from how we defined . Also, from how we defined a and b.) This gives the conclusion that exists so is differentiable at as required.
It is an oddly phrased question though
MVT states that such that , but if defines a single point, then we can clearly have that .
I was thinking, and I don't know if this also holds, but we know by MVT that . For any other point such that , is there guaranteed a subinterval such that ?
Sorry if this seems jumbled a bit, but I greatly appreciate all the help.
oh, I thought was a generic point on .
What I wrote still holds, turn the in my proof to a and show that it's continuous and differentiable for . Therefore can equal so must be differentiable at .
(If every is continuous and differentiable where then if then must be must continuous and f must be differentiable at ).
This is incorrect. can only be used for sets but c is a point.