Is the conjugate of a complex function holomorphic?

Hello

Suppose is holomorphic on the open disc of the complex plane.

How would one go about proving is holomorphic in ?

I attepted this by writing out the definition of the derivative, substituting doing the algebra and using the facts that the real and imaginery parts of are differentiable and if then

Is this the way this question would be tackled?? i'm not sure if my proof is a hundred percent correct.

Also I want to prove is differentiable at iff f'(a) = 0.

I'm a bit stuck on this part. Any ideas?

Incidently this is question 5.11 from Introduction to Complex Analysis by Priestly.