Let $S\subset \mathbb{C}$ be a simply connected domain (i.e. every point in complement of $S$ can be connected to $\infty$). Let $C=\{z:|z-\alpha|=r\}$. Would you help me to show that for all $z\in S$, $\delta(z)=\sup \{u:D(z,u)\subset S\}$ is continuous function?
Continous function defined by supremum of Radius of disc
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complex-analysis