Nowhere differentiable functions with respect to the position

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{C}$ such that $\partial \Omega$ does not contain isolated points. Let $R(\Omega)$ be the space of uniform limits on $\overline{\Omega}$ of rational functions with poles off $\overline{\Omega}$, endowed with the supremum norm. We prove that either generically all functions $f$ in $R(\Omega)$ satisfy % $$ \limsup_{\substack{z \to z_0 z \in \partial \Omega}} \Big| \frac{f(z) - f(z_0)}{z - z_0} \Big| = + \infty $$ for every $z_0 \in \partial \Omega$ or no such function in $R(\Omega)$ meets this requirement. In the first case, the generic function $f \in R(\Omega)$ is nowhere differentiable on $\partial \Omega$ with respect to the position. We give specific examples where each case of the previous dichotomy holds. We also extend the previous result to unbounded domains.