Jordan curve and Conformal maps

Question

Let $\mathbb D$ be the unitary open disk, $D$ a bounded domain(open and connected) with boundary a jordan curve and $f$ a conformal map from $\mathbb D$ to $D$, is it true that we can extend $f$ as an homeomorphism from $\overline{\mathbb D}$ to $\overline{D}$? and if it is true, does it hold for two bounded domains with boundary a jordan curve?

Answer 1

Yes to the first question - this is a theorem of Caratheodory.

And this implies a yes to the second question: If $f_j:\overline{\Bbb D}\to \overline D_j$ for $j=1,2$ are as above then $f_2\circ f_1^{-1}:\overline D_1\to\overline D_2$.