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!!Without regard for computability!!
One classic criterion is this. $(U,f)$ analytically continues to $(V,g)$ iff there is a finite sequence $(U_k,f_k), k=0,\dots,n$ of analytic elements, with $(U_0,f_0) = (U,f)$ and $(U_n,g_n) = (V,g)$ where, for $k=1,\dots,n$, we have $U_{k-1}\cap U_k \ne \varnothing$ and $f_{k-1}(z) = f_k(z)$ for all $z \in U_{k-1}\cap U_k$.