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 $\mathbb D$?
> 
> 
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Riemann mapping theorem and smoothness on the boundary
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).


