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.