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 $\mathbb D$?