Let $G$ be an open connected set and let $D \subset G$ be a dense set. Let $(f_n)$ be a sequence of holomorphic functions in $G$ and assume $f_n \rightarrow 0$ pointwisely on $D$. Can we deduce that $f_n$ converges pointwisely to a function $f$?
If this is true, by Osgood theorem (Convergence of a sequence holomorphic functions) $f_n$ converges uniformly in a dense open set $D'$ of $G$, so we deduce $f$ is holomorphic in $D'$. Since $f\mid_D = 0$ it will follow that $f\mid_{D'} = 0.$
Now, assume we do not know that $f_n$ converges pointwisely but instead $f_n$ is locally bounded. By Vitali-Porter theorem (https://mathoverflow.net/questions/82787/vitalis-theorem-on-convergence-of-holomorphic-functions) $f_n$ converges uniformly on compacts subsets of $G$ to an analytic function, but it only 'needs' $D$ to have an accumulation point.
My question is the following: could we deduce something like Vitali or Osgood theorem using only pointwise convergence on a dense set $D$?
Thank you very much!
Take $g_n(z) = \prod_{i=1}^n (z-r_i)$ for $r_i$ some enumeration of rational points in $\mathbb{C}$ and let $c_n = n / g_n(\pi)$. Then $f_n(z) = c_n g_n(z) \rightarrow 0$ for any rational $z$ - the sequence is eventually $0$. But $f_n(\pi) = n \rightarrow \infty$.