You're overcomplicating it! Note that
Also, I think that the proper term is 'differentiable', not 'derivable'.
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!
Another person who wants us to "bare" with him! This website is getting too raw for me!
liquidFuzz, in order that a function be differentiable at z= a, you must have exist. Since that denominator obviously goes to 0, in order that the limit exist, the numerator must also exist (a necessary though not sufficient condition). That is, we must have , whence . Letting that is the same as .