Constantin Carathéodory gave another proof of the theorem in 1912, which was the first to rely purely on the methods of function theory rather than potential theory., citing His proof used Montel's concept of normal families, which became the standard method of proof in textbooks. Carathéodory continued in 1913 by resolving the additional question of whether the Riemann mapping between the domains can be extended to a homeomorphism of the boundaries (see Carathéodory's theorem).
