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