I'm learning about analytic continuations and global analytic functions which were seen to be connected components of the sheaf of analytic germs.
Sometimes we get problem sets in which we are asked to define the largest branch possible of some function $f$, where $f$ satisfies some functional relationship. For example, $f^2(z) - z\sin(z) = 0$
Answers to such problem sets may tell us to "note that $f$ has a branch point at such and such location." Do statements like that even make sense? What if two different global analytic functions (two components of the sheaf) can satisfy this relation and have completely different branch points? Don't we have to know that a functional relation $G(z, f) = 0$ specifies a unique component in the sheaf?
That could happen: the equation $$(f(z)^2-z)(f(z)-z)=0$$ is satisfied by a function with a branch point at $0$, and also by an entire function.
However, we may be able to say something branch points for some equations. Let's consider equations of the form $\Phi(f(z)) = \Psi(z)$. Suppose that $\Psi'(z_0)\ne 0$ and $\Phi'$ vanishes at every point of the set $\{z: \Phi(z) = \Psi(z_0)\}$. Then $f$ cannot be holomorphic at $z_0$. This consideration applies, e.g., to $f(z)^2=z$.