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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 $\partial U \to S^1$ at the boundary.

To obtain the 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 $U$.

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 $\partial U \to S^1$ 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 $U$.

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 $\partial U \to S^1$ at the boundary.

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 $U$.

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 $\partial U \to S^1$ at the boundary.

To obtain the 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 $U$.

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 $\partial U \to S^1$ at the boundary.

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 $U$.

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 $\partial U \to S^1$ at the boundary.

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 $U$.