Prove that if is derivable in then is continuous in .

I started with the definition.

Using Cauchys integral formula.

=

=

=

Now comparing the result with Cauchy's Integral formula, first order.

=

What happens with the right side if ?

I'm sort of in the dark here... I thought maybe I proved something by showing how Cauchy's and the definition of derivation dance so nicely together.

Help or any pointers would be appreciated!