Holomorphic function

Hey who can help me with this:
Let f be holomorphic in a neighborhood of a closed disk . Then is a holomorphic function on .
Why, and which one?

Did you compute for ?

Why do I do this, to check whether its holomorphic?

Yes, we try to check the definition of holomorphic.

Im somehow lost, when I compute the quotient I just get f'(z) multiplied by a constant. And I dont really use the fact that z is outside D.