Analytic continuation and singularities

Question

Suppose that $z\in\mathbb{C}$ and $f$ is a holomorphism on $U$ such that there is no holomorphism $g$ extending $f$ on $V\supseteq U$ such that $z\in V$. Is it possible for there to be such a $g$ and a holomorphism $h$ on $W$ with $V\cap W\neq\emptyset$ such that $g|_{V\cap W}=h|_{V\cap W}$ with $z\in W$? That is, are singularities preserved by analytic continuation?

Answer 1

The answer to radius of convergence with Taylor series, functions with branch cuts is related, and indeed its negative answer can be easily seen to imply a negative answer to this question. The global analytic function $\frac{1}{\log(z)}$ is given as a counterexample. Since $\log(z)=0$ in only one branch, there is a singularity at $0$, but also a disappearing one.