Continuous Maps Admitting No Tangent Lines: A Centennial of Besicovitch Functions

Abstract

One of the most influential examples in analysis is a Weierstrass function from to that is continuous but differentiable at no point. However this map, as well most of the others among the myriad similar examples, still admits vertical tangent lines. The examples of continuous maps that admit no tangent line in any direction are also known; however, all currently existing presentations of such maps are not easily accessible due to their very complicated descriptions and hard-to-follow proofs of their desired properties. The goal of this article is to present in an accessible way two such examples. The first—a coordinate of the classical Peano space-filling curve—is simpler, but admits one-sided vertical tangent lines at some points. The second is a variation of a function from a 1924 paper of Besicovitch, which is continuous but admits no one-sided tangent line in any direction. The proofs of nondifferentiability of these two examples will be facilitated by a simple yet general lemma that also implies nondifferentiability of other similar maps, including those of Takagi and van der Waerden.