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What happens about the pointwise limit of a sequence of analytic function defined on a domain $D\subseteq \Bbb C$. For example can we say that it is also analytic or it is continuous?

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Osgood's theorem says that if $(f_n)$ is a sequence of holomorphic functions, converging pointwise $f$ on $D$, then we can find an open, dense set $\Omega \subset D$ such that $f$ is holomorphic on $\Omega$, and in fact the convergence is locally uniform on $\Omega$.

In general though, $\Omega$ is a proper subset of $D$. See Daniel Fischer's link in the comment, or this question for examples.

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