an example for sheaves

Question

I want to know whether we can define a sheaf for holomorphic functions or not,I check the axioms of when a presheaf could be a sheaf,but I'm not sure it is correct.for axiom 1 I use identity theorem and for axiom 2 we should say there exist a holomorphic function on an open set U such that restriction of f to Ui is fi. I suppose f be union of fi and because of our hypothesis we conclude that fi=fj every where and that union will be fi,is this a correct solution?

Answer 1

The answers is yes, in fact the sheaf of holomorphic functions is one of the classical examples.

The fact that it is a presheaf is easy as you mentioned.

You have use correctly the identity theorem.

To the second axiom you should demonstrate that your f is in fact holomorphic to sum up. But you are in the correct way (In $\mathbb{R}$ with infinite class functions you have the mix-function theorem just follow a similar arguent)