Topological genericity of nowhere differentiable functions in the disc and polydisc algebras

Abstract

In this paper we examine functions in the disc algebra $\mathcal{A}(D)$ and the polydisc algebra $\mathcal{A}(D^I)$, where $I$ is a finite or countably infinite set. We prove that, generically, for every $f\in \mathcal{A}(D)$ the continuous periodic functions $u=Ref{|}_{\mathbb{T}}$ and $\tilde{u}=Imf{|}_{\mathbb{T}}$ are nowhere differentiable on the unit circle $\mathbb{T}$. Afterwards, we generalize this result by proving that, generically, for every $f\in \mathcal{A}(D^I)$, where $I$ is as above, the continuous periodic functions $u=Ref{|}_{{\mathbb{T}}^I}$ and $\tilde{u}=Imf{|}_{{\mathbb{T}}^I}$ have no directional derivatives at any point of ${\mathbb{T}}^I$ and every direction $v\in {\mathbb{R}}^I$ with $\Vert v{\Vert }_{\infty }=1$.