Functional Equations and Combinatorial Constructions of Pathological Functions in Real Analysis

This paper explores the construction of continuous nowhere differentiable functions, a cornerstone in real analysis since Weierstrass’ 1872 example. Two frameworks are developed and analyzed rigorously: (1) a functional-equation approach leading to generalized Weierstrass-type functions, and (2) a combinatorial construction based on binary digit expansions. Existence, uniqueness, continuity, and nowhere differentiability are proved in detail. The paper highlights how simple recursive or symbolic rules generate analytic irregularity, with potential applications to harmonic analysis, fractal geometry, and dynamical systems.