A disk chain is a sequence of open disks $(C_0,\dots ,C_n)$ with centers $(c_0,\dots ,c_n)$ such that $c_{k-1},c_k\in C_{k-1}\bigcap C_k$. We say that $f_n:C_n\to \mathbb{C}$ is an analytic continuation of a holomorphic function $f_0:C_0\to \mathbb{C}$ along the chain $(C_0,\dots ,C_n)$ if there 1;;exists $f_k:C_k\to \mathbb{C}$ holomorphic such that $f_k{|}_{C_{k-1}\bigcap C_k}=f_{k-1}{|}_{C_{k-1}\bigcap C_k}$. Notice further that these maps are unique, by the uniqueness theorem.