Carathéodory's theorem on boundary extension

Remarks

Such an $f$ exists by the Riemann mapping theorem, since $\Omega$ is simply connected by the Jordan curve theorem. The proof rests on a length–area estimate. Let $\zeta$ be a point of the unit circle and let $r$ be small. By the Cauchy–Schwarz inequality and the finite area of $\Omega$, some arc $\set{\abs{z - \zeta} = \rho} \cap \mathbb{D}$ with $r^2 < \rho < r$ has an image of length at most $C/\sqrt{\log(1/r)}$. Because $J$ is a Jordan curve, a short crosscut of $\Omega$ cuts off a piece of small diameter, and this gives the continuity of $f$ at $\zeta$. The Jordan property also gives injectivity on the circle. The converse holds: if $f$ extends to such a homeomorphism, the boundary of $\Omega$ is the image of the circle, hence a Jordan curve. A more general form says that $f$ extends continuously to $\overline{\mathbb{D}}$ if and only if the boundary of $\Omega$ is locally connected. Example: the boundary of the slit disc $\mathbb{D} \setminus [0, 1)$ is not a Jordan curve. A conformal map from $\mathbb{D}$ onto it extends continuously, but two points of the circle go to each inner point of the slit, so the extension is not injective.

Going deeper

For a student

The theorem lets one transfer boundary value problems, such as the Dirichlet problem, from the disc to any Jordan domain. It also shows that a conformal map between two Jordan domains extends to a homeomorphism of their closures, by composing two Riemann maps. For domains with worse boundaries, the right replacement for boundary points is the notion of a prime end. Carathéodory's prime end theorem gives a bijection between the circle and the prime ends.