# Nowhere differentiable functions with respect to the position

Source: [Nowhere differentiable functions with respect to the position](<https://www.sota2.com/research/paper/68fa53961e209131ce9848bd>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T12:32:14.678Z

Authors: K. Kavvadias, K. Makridis

Published: 2017-01-17T21:32:53.000Z

## 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.

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1701.04875>)

[PDF](<https://arxiv.org/pdf/1701.04875.pdf>)