One can invoke Carathéodory's theorem.
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 $U$.