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# Find the derivative of the given piecewise function

Let

Then define

1. Prove

2. Prove that is differentiable at 0 and compute .

1. Proof. Let be given. Then choose . Then if

we have

Hence, we have found a such that whenever . Therefore,

2. 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

### One comment

1. Mihajlo says:

There is probably a typo in part a), because $|\delta| < \epsilon$, and not |h|=epsilon.