Topological genericity of nowhere differentiable functions in the disc algebra

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.