Riemann mapping theorem and smoothness on the boundary Let $U\subset \mathbb C$ be open, bounded, simply connected, with $C^\infty$ boundary. Apply the Riemann mapping theorem to get a bilolomorphic isomorphism
$$
f:U\to \mathbb D
$$
between $U$ and the unit disc $\mathbb D:=\{z\in \mathbb C:|z|<1\}$.

How can I see that $f$ extends to a $C^\infty$ map from the closure of $U$ to the closure of $\mathbb D$?

 A: See http://maths.sogang.ac.kr/shcho/pdf/P18.pdf, theorem 1.3 (which is the same as theorem 3.4).
A: Another answer, since it is different from the previous one:
The result you want was proved by Painleve in 1887, long BEFORE Cartheodory's theorem. The proof is given in the very nice survey article:
http://www.ams.org/journals/bull/1990-22-02/S0273-0979-1990-15879-3/S0273-0979-1990-15879-3.pdf (page 238). The paper is:
S. Bell
MAPPING PROBLEMS IN COMPLEX ANALYSIS AND THE D-BAR problem (bull AMS, 1990). (it still uses elliptic regularity, but he refers to other proofs which are different).
A: This is well-known result by Kellogg  (O. Kellogg: On the derivatives of harmonic functions on the boundary, Trans. Amer.Math. Soc. 33 (1931), 689-692.), and  Warschawski (On the higher derivatives at the boundary in conformal mapping,} Trans. Amer. Math. Soc, {\bf 38}, No. 2 (1935), 310-340.), where they prove even more, that the if the boundary is C^{n,\alpha}, then the conformal parametrization is C^{n,\alpha} up to the boundary.
A: By a theorem of Caratheodory it extends to a homeomorphism.
