Continuous extension of conformal maps

Abstract

For a simply connected domain $G$, let $\partial_{a}G$ be the set of accessible points in $\partial G$ and let $\partial_{n} G=\partial G-\partial_{a}G$. A point $a\in\partial G$ is called semi-unreachable if there is a crosscut $J$ of $G$ and domains $U$ and $V$ such that $G-J=U\cup V$ and $a\in(\partial_{n} U\cup\partial_{n} V)-J$. We use $\partial_{sn}G$ to denote the set of semi-unreachable points. In this article we show that a univalent analytic function $\psi$ from the unit disk $D$ onto $G$ extends continuously to $\overline D$ if and only if $\partial_{sn}G=\emptyset$. As a consequence, we provide a very short and elementary proof for the Osgood conjecture: if $G$ is a Jordan domain, then $\psi^{-1}$, the Riemann map, extends to be a homeomorphism from $\overline G$ to $\overline D$.