Prove this derivative is continuous
Let be a neighborhood of 0 in .
Suppose that is twice differentiable , f' and f'' are continuous, and .
Define by if and if
Show that g' exists and is continuous on U.
Proof so far.
Claim: g' exists on U.
By the division theorem, we know that , implies that for , we would have , so we have .
For , we have
Claim: g' is continuous on U.
Write , so we have
In the case that :
Given , pick , then for each and , we have
I have a feeling I'm very wrong as I'm stuck at this point, any hints? Thank you.