Let
Then define
- Prove
- Prove that is differentiable at 0 and compute .
- Proof. Let be given. Then choose . Then if
we have
Hence, we have found a such that whenever . Therefore,
- Proof. To show has a derivative at 0 we must show that the limit
exists. We note that (since 0 is rational and ), and compute
There is probably a typo in part a), because $|\delta| < \epsilon$, and not |h|=epsilon.