Carathéodory-Torhorst theorem An extension of the theorem states that a conformal isomorphism , where is a simply connected subset of the Riemann sphere, extends continuously to the unit circle if and only if the boundary of is locally connected. This result is often also attributed to Carathéodory, but was first stated and proved by Marie Torhorst in her 1918 thesis, under the supervision of Hans Hahn, using Carathéodory's theory of prime ends. More precisely, Torhorst proved that local connectivity is equivalent to the domain having only prime ends of the first kind. By the theory of prime ends, the latter property, in turn, is equivalent to having a continuous extension. Notes References Conformal mappings Homeomorphisms Theorems in complex analysis