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If $\Omega_1$ and $\Omega_2$ are two nonempty disjoint open subsets ${\bf C}$ and $\{f_n\}$ is a sequence of analytic functions from $\Omega_1 \to \Omega_2$ which converges pointwise to a function $f : \Omega \to {\bf C}$, then $f$ is analytic and $f(\Omega_1) \subset \Omega_2$ unless $f$ is constant.

I think the idea is to use Montel's theorem but the condition is not satisfied.

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Hint: Let $a\in \Omega_1.$ Then the functions $\frac{1}{f_n-a}$ are uniformly bounded on $\Omega_1.$

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