Carmen Pérez Quintana: När kontinuitet inte räcker: Weierstrassfunktionen och deriverbarhetens gränser

Abstract: During the 19th century, it was widely believed that continuous functions were, for the most part, also differentiable. This view was challenged in 1872 by Karl Weierstrass through the construction of a function that is continuous everywhere but differentiable nowhere. This work investigates the relationship between continuity and differentiability through an analytical study of the Weierstrass function and a modern formulation of the function and a modern reformulation of it. A theoretical framework for uniform convergence is established to demonstrate the function’s global continuity. Subsequently, a modern formulation of the function based on a piecewise linear sawtooth function, where its difference quotient is studied to show that a finite limit is absent at every point. The thesis highlights how this construction challenges intuitive assumptions about the behavior of functions and emphasizes the importance of mathematical rigor in real analysis.