Topological genericity of nowhere differentiable functions in the disc algebra

Abstract

In this paper we introduce a class of functions contained in the disc algebra ${\mathcal{A}(D)}$ . We study functions ${f \in \mathcal{A}(D)}$ which have the property that the continuous periodic function ${u = {\rm Re}f|_{\mathbb{T}}}$ , where ${\mathbb{T}}$ is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function ${f \in \mathcal{A}(D)}$ has the above property. Afterwards, we strengthen this result by proving that, generically, for every function ${f \in \mathcal{A}(D)}$ , both continuous periodic functions ${u = {\rm Re}f|_\mathbb{T}}$ and ${\tilde{u} = {\rm Im}f|_\mathbb{T}}$ are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire’s Category Theorem.