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

> If $U$ is a simply connected open
> subset of the complex plane with a
> [Jordan curve][2] as boundary then the
> [Riemann map][3] $f : U \to \mathbb D$
> extends continuosly to the boundary
> and the extension is a homeomorphism
> $\partial U \to S^1$ at the boundary.

To obtain the sought 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]: https://en.wikipedia.org/wiki/Carathéodory%27s_theorem_(conformal_mapping)
  [2]: https://en.wikipedia.org/wiki/Jordan_curve_theorem
  [3]: https://en.wikipedia.org/wiki/Riemann_mapping_theorem