Topological genericity of nowhere differentiable functions in the disc algebra (Q403041)

Denote by \(\mathcal{A}(D)\) the Banach algebra of all holomorphic functions on the open unit disk \(D\) that extend continuously to its closure. The main result of this paper is the following theorem. {Theorem.} Let \(E\) be the class of all functions \(f\in \mathcal{A}(D)\) such that the function \(u = \mathrm{Re} f |_{\mathbb{T}}\) is not differentiable at any point \(\theta\in \mathbb R\). Then \(E\) is residual in \(\mathcal{A}(D)\), i.e., it contains a dense \(G_\delta\)-subset of \(\mathcal{A}(D)\).