TITLE: Does pointwise convergence of holomorphic functions on the boundary imply pointwise convergence in the interior?
QUESTION [5 upvotes]: Let $\Omega$ be a simply connected open set in the complex plane and $\gamma$ be a simple path inside $\Omega$. Suppose $f_n$ is a sequence of holomorphic functions converging pointwise to 0 on $\gamma$. Does it imply that $f_n$ converges pointwise on the region enclosed by $\gamma$?

REPLY [10 votes]: For a counterexample, let $\gamma$ be the unit circle.  Let $$A_n = \{z \in \gamma:\; \text{Im}(z) \in [-1,0] \cup [1/n, 1]\}$$  By Runge's theorem there is a polynomial $f_n$ such that $|f_n| < 1/n$ on $A_n$ but $f_n(0) = (-1)^n$.  We then have
$f_n \to 0$ pointwise on $\gamma$ but $f_n(0)$ does not converge.