O floco de neve de Koch e suas propriedades: funções contínuas sem derivada em ponto algum

Abstract

In this work, we present the existence of real continuous functions that have no derivative at any point. For this, we use the function developed by the mathematician Helge von Koch as an example, demonstrating that this function is continuous at all points but not differentiable at any point. We show how this curve is constructed and discuss itsproperties. To highlight these facts, many constructions of such functions are based on infinite series of functions. Therefore, we introduce some fundamental concepts and results from Mathematical Analysis, specifically, Sequences and Series of functions, which allow us to investigate the continuity and differentiability properties. Finally, we will comment on an interesting result that reveals that the set of these functions constitutes a dense and residual set in the complete metric space, meaning that these functions exist abundantly. The proof of this statement is based on Baire’s Theorem, which generally states that any countable union of thin sets is so small that its complement is dense.