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 % $$\underset{\begin{array}{c}z\to z_{0}z\in \partial \Omega \end{array}}{\mathrm{limsup}}\left|\frac{f(z)-f(z_0)}{z-z_0}\right|=+\infty$$ 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.