Topological genericity of nowhere differentiable functions in the disc algebra

TL;DR

This paper demonstrates that in the disc algebra, it is generically typical for the boundary functions of most functions to be nowhere differentiable, highlighting a topological genericity of such irregular boundary behavior.

Contribution

The paper proves that the set of functions in the disc algebra with nowhere differentiable boundary functions is generic, using Baire's Category Theorem without relying on classical examples like the Weierstrass function.

Findings

Most functions in the disc algebra have boundary functions that are nowhere differentiable.

The class of functions with nowhere differentiable boundary functions is non-empty and generic.

Both real and imaginary parts of boundary functions are generically nowhere differentiable.

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 $u = Ref|_{\mathbb{T}}$, 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 $u=Ref|_\mathbb{T}$ and $\tilde{u} = Imf|_\mathbb{T}$ are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire's Category Theorem.