To Generalize Carath\'eodory's Continuity Theorem

Let $\varphi: D\rightarrow \Omega$ be a homeomorphism from a circle domain $D$ onto a domain $\Omega\subset\hat{\mathbb{C}}$. We obtain necessary and sufficient conditions (1) for $\varphi$ to have a continuous extension to the closure $\overline{D}$ and (2) for such an extension to be injective. Further assume that $\varphi$ is conformal and that $\partial\Omega$ has at most countably many non-degenerate components $\{P_n\}$ whose diameters have a finite sum $\displaystyle\sum_n{\rm diam}(P_n)0$ at most finitely many of its components are of diameter greater than $C$.