Carathéodory's theorem (conformal mapping)

In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, published by Carathéodory in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve.

Source: Wikipedia — Carathéodory's theorem (conformal mapping) (CC BY-SA 4.0)