Nowhere differentiable functions of analytic type on products of finitely connected planar domains (Q667814)

A generalization of the Banach-Mazurkiewicz theorem on nowhere differentiable functions to the case of functions defined on a bounded multiply connected domains in the complex plane is given. The main result has the following form. Theorem. Let $\Omega\in\mathbb{C}$ be a multiply connected domain bounded by a finite number of disjoint Jordan curves. Let $V_0, V_1, \dots, V_{n-1}$ be connected components of $\widehat{\mathbb{C}}\setminus \Omega$, $\infty\in V_0$ and $\Omega_0 = \widehat{\mathbb{C}}\setminus V_0$, $\Omega_1 = \widehat{\mathbb{C}}\setminus V_1,\dots,\Omega_{n-1} = \widehat{\mathbb{C}}\setminus V_{n-1}$. Denote by $\mathcal{A}(\Omega)$ a set of all functions $f: \overline{\Omega} \rightarrow\mathbb{C}$ holomorphic in $\Omega$ and continuous in $\overline{\Omega}$ and by $\phi_j : \overline{\mathbb{D}} \rightarrow \overline{\Omega}_j$ the Riemann maps of the closed unit disc onto $\overline{\Omega}_j$. The class of functions $f\in\mathcal{A}(\Omega)$ such that for every $j=0, 1, \dots, n-1,$ the composition $\left(f \circ \phi_j\right)\Bigl|_{\mathbb{T}}$ is nowhere differentiable is dense and $G_{\delta}$ in $\mathcal{A}(\Omega)$. A generalization of the above theorem to the case of infinitely connected domains is presented, too.