Global theory of holomorphic functions I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$.  Recently, I was looking at the last chapter of Lars Ahlfors Complex Analysis book and I discovered striking similarities between my approach and that of Weierstrass.
First, lets start with a definition of holomorphy.  Lets say that a complex valued function $f(z)$ is holomorphic at $z_0$ if $\lim_{z \to z_0}\frac{f(z)-f(z_0)}{z-z_0}$ exists and is finite.  So holomorphy is implicitly a local property that not only involves a single point $z_0$ but all of the points in a neighborhood containing $z_0$.
Now let us proceed and try to develop a global theory of holomorphic functions.  The idea is to fix a point $z_0$ in the complex plane and look at the behavior of a particular holomorphic function $f(z)$ in a neighbored $B(z_0, r)$.  It will become more clear why I take this approach.  Now we want to extend the domain of holomorphy of this fixed function $f(z)$ defined initially in an open neighborhood of $z_0$.  One way to do this is to use an equivalence relation.
We say that $f(z)$ R $g(z)$ if $f(z) = g(z)$ for all $z$ in any open set containing $z_0$.  This is clearly an equivalence relation which partitions the set of holomorphic extensions of $f(z)$ near $z_0$.  By choosing a representative $g$ of each equivalence class we see that there exists open sets $U_g$ (the domain of $g$) such that $f(z) = g(z)$ on $B(z_0, r) \cap U_g$ and $g$ extends $f$ to $B(z_0, r) \cup U_g$.  Now let $U_{z_0} = \cup_g U_g$ where the union is taken over all equivalence classes described above.  Then $U_{z_0}$ is the largest open set where we can find a holomorphic extension of $f(z)$ at $z_0$.  That is, there exists a global extension $G(z)$ of $f(z)$ at $z_0$ in the sense that $G$ is holomorphic and the domain of $G$ is $U_{z_0}$.
Questions and Comments:

*

*Is this formulation of analytic continuation correct and is it any different from Weierstrass' approach?


*I cannot prove existence and holomorphy of $G$ but I suspect that follows from a simple relation between the representatives of each equivalence class and $G$.


*How to I pass from a fixed point $z_0$ to the entire domain of $f$?  If $D$ is a countable union of open sets then I think its possible.
 A: This is more a long comment than an answer. Despite the clean and compact counterexample to the "global" theory of analytic continuation proposed by GH from MO in their answer to the question, a similar theory, more refined and possibly more successful, was proposed long ago by Gaetano Fichera in [1].
Fichera ([1], p. 214) starts by considering a metric space $M$, a general set $S$, the basis for the topology of $M$ made by open balls $B_{x,\rho}\triangleq B(x,\rho)$, for a given $x\in X$ and $0<\rho\le+\infty$ and the set of couples $(B_{x,\rho}, f)$ (called functional elements) where $f:B_{x,\rho}\to S$ is a $S$-valued function. A Weirstrass class $\mathscr{W}$ of functional elements is a class of functional elements verifying the following axioms:

*

*For any two functional elements $(B_{x_1,\rho_1}, f_1)$ and $(B_{x_2,\rho_2}, f_2)$ such that $B_{x_1,\rho_1}\cap B_{x_2,\rho_2}\neq \emptyset$ and $f_1=f_2$ in an open set contained in $B_{x_1,\rho_1}\cap B_{x_2,\rho_2}$, then
$$
f_1(x)=f_2(x) \quad \forall x\in B_{x_1,\rho_1}\cap B_{x_2,\rho_2}.
$$

*If $(B_{x_1,\rho_1}, f)\in \mathscr{W}$ then, for any open ball $B_{x_2,\rho_2}, \subset B_{x_1,\rho_1}$, also $(B_{x_2,\rho_2}, f)\in \mathscr{W}$.

After this, he gives respectively a definition of a maximal functional element respect to a (Weierstrass) class $\mathscr{W}$ and of a complete Weierstrass class, develops his "abstract" approach to analytic continuation and gives several examples of applications, namely to the global inversion of locally invertible continuous maps ([1], §6, pp. 222-224), to the analytic continuation of functions defined on a differentiable manifold ([1], §7, pp. 224-225) and to the continuation of solutions to systems of differential equations (involving total differentials) ([1], §7, pp. 226-227).
Notes

*

*I tried to update a bit Fichera's notation by putting $\Sigma=M$, $\mathscr{C}=\mathscr{W}$ and $I(x,\rho)=B(x,\rho)$ in order to clarify my exposition.

*The filter structure a la Cartan used in the approach is evident: however, as Fichera notes ([1], footnote 2, p. 214), $\mathscr{W}$ has not a presheaf structure.

Reference
[1] Gaetano Fichera, "Teoria astratta del prolungamento di Weierstrass e applicazioni",  (Italian) Annali di Matematica Pura ed Applicata, IV Serie, 49, 213-227 (1960), DOI 10.1007/BF02414051, MR0117720, Zbl 0095.06001.
A: In general, the holomorphic extension to $U_{z_0}$ that you envision does not exist. Take, for example, the principal branch of the logarithm on the disk $B(1,1)$. It has a holomorphic extension to $\mathbb{C}\setminus i[0,\infty)$, and also a holomorphic extension to $\mathbb{C}\setminus -i[0,\infty)$. Yet, it has no holomorphic extension to $\mathbb{C}\setminus\{0\}$, which is the union of the mentioned two domains.
