$f(z)$ is analytic on the open unit disk $\{|z| < 1 \}$. Assume that $\lim_{z\to t} f(z) = c$ for all $t$ on an arc $\{e^{i\theta}: \theta \in (a,b)\}$ and some constant $c$. How to proof $f(z)$ is constant on the open unit disk?
I have read solutions to this question: function constant on arc is constant on boundary, but it assume $f(z)$ is continuous up to the boundary of the disk. Without the condition, we cannot use Cauchy integral theorem for $f(z)$, and I have no idea how I can associate the boundary with the interior.
