One can invoke [Carathéodory's theorem][1]. 

> If $U$ is a simply connected open
> subset of the complex plane with a
> Jordan curve as boundary then the
> Riemann map $f : U \to \mathbb D$
> extends continuosly to the boundary
> and the extension is a homeomorphism
> $\partial U \to S^1$.

To obtain to sougth function, it suffices to consider a simply connected open set $U\subset \mathbb C$ having  a nowhere analytic  Jordan curve as boundary and take the inverse of the Riemann map of $U$.  

  [1]: http://en.wikipedia.org/wiki/Carath%25C3%25A9odory's_theorem_(conformal_mapping)