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?
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.