the boundary correspondence
The Riemann mapping theorem builds a conformal map between the INTERIORS of two regions, and says nothing on its own about what happens at the edges. But for applications — solving a boundary-value problem, matching data along the rim — you desperately want the map to extend continuously to the boundary, so that points on the edge of your domain correspond cleanly to points on the unit circle. The boundary correspondence is the study of when, and how nicely, the interior conformal map reaches all the way out to the boundary.
The interior map can behave badly at the boundary if the boundary itself is wild: for a domain whose boundary is a fractal or has inward spikes and slits, the map may not extend to a single-valued continuous function on the closure at all (different paths approaching one boundary point can land at different circle points). The clean positive result is for Jordan domains. A Jordan curve is a simple closed curve — a loop that does not cross itself — and a Jordan domain is the region it bounds. Carathéodory's theorem says exactly that for a Jordan domain the Riemann map extends to a homeomorphism between the closed domain and the closed disk: the boundary curve corresponds continuously and bijectively to the unit circle, preserving the cyclic order of points. Smoother boundaries give smoother correspondence: if the boundary is a smooth (or analytic) curve, the map and its derivatives extend smoothly to the boundary too.
This is what makes conformal mapping usable in physics and engineering. To solve Laplace's equation (heat, electrostatics, ideal flow) on a complicated Jordan domain with prescribed boundary values, you transport the boundary data to the unit circle through the boundary correspondence, solve the easy disk problem (Poisson integral), and map the answer back. Honest caveats: the homeomorphism conclusion needs the boundary to be a Jordan curve; for non-Jordan boundaries one must pass to the subtler theory of prime ends to make sense of 'boundary points'. And continuous extension does not by itself give a smooth or conformal extension at corners — at a boundary corner of interior angle alpha pi the map behaves like a power and its derivative may blow up or vanish there.
Map the open upper half-disk (a Jordan domain bounded by a semicircle and a diameter) conformally onto the unit disk. By Carathéodory's theorem the map extends to a homeomorphism of the closures: each point of the boundary — the curved arc and the flat segment, including the two corners — corresponds to a unique point of the unit circle, in the same going-around order.
For a Jordan domain the boundary maps homeomorphically onto the circle, corners and all.
Continuous boundary extension requires the boundary to be a Jordan curve (Carathéodory); for wilder boundaries you need prime ends. Continuity is not smoothness — at a corner the derivative can blow up or vanish even though the map extends continuously.