Anayltic Continuation on complement of countable set

Question

I'm reading Reconstructing a neural net from its output, and there's this remark.

Let $F_1$ and $F_2$ be analytic on $\Omega_1, \Omega_2$, with $\Omega_1$ connected and $\mathbb{C} \setminus \Omega_2$ countable. Say $F_1= F_2$ to infinite order at some point in $\Omega_1 \cap \Omega_2$. Then $F_1 = F_2$ on $\Omega_1 \cap Omega_2$.

I do know that by identity theorem, if $F_1 = F_2$ on some $S \subseteq \Omega_1 \cap \Omega_2$, then $F_1 = F_2$. However, I don't know what is meant by "infinite order", so I don't know what the remark is saying.

Thanks!