Generalized continuous nowhere differentiable function families

The existence of continuous nowhere-differentiable functions is a fundamental phenomenon in real analysis, challenging the classical intuition that continuity generally implies differentiability. Once regarded as pathological constructions, such functions are now recognized as abundant in the space of continuous functions, both in the sense of Baire category and through notions of prevalence. In this paper, we develop a general framework for constructing continuous nowhere differentiable functions represented by trigonometric series with decaying amplitudes and rapidly increasing frequencies. We establish a sufficient condition for nowhere differentiability that applies to a broad class of Fourier-type series, thereby providing a unified criterion that encompasses several classical constructions, including the Weierstrass function and related examples. The proposed approach also yields new families of continuous nowhere differentiable functions under flexible conditions on the amplitudes and frequencies. These results contribute to a systematic understanding of the mechanisms underlying the breakdown of differentiability and further illustrate the richness and ubiquity of nowhere-differentiable behavior within the class of continuous functions.