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 $\mathbb{R}$ to $\mathbb{R}$ 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.

MSC::

• Primary 26A27
• Secondary 26A30
• 26A24

Notes

1 The first woman who received a doctorate in any field in Germany, degree granted in 1895.

2 More specifically, the function from [39] is just a different description of the original Besicovitch function $B_{1/2}$.

3 One of the pioneering women in American mathematics; see [18].

4 This seems to be a reason for an incorrect interpretation of the function B₁ in a 2020 paper [16] of Serge Dubuc leading to a conclusion that the Besicovitch function B₁ admits one-sided infinite derivatives (while the claim is true only for the function that was understood as function B₁).

5 However, the function need not be affine-similar for the lemma to be useful, as we see in its use in Section 4.

6 We define the support of an $f:X\to \mathbb{R}$ as the set $f^{-1}(\mathbb{R}\setminus \{0\})$ rather than its closure.

Additional information

Notes on contributors

Krzysztof Chris Ciesielski

KRZYSZTOF CHRIS CIESIELSKI received his Master’s and Ph.D. degrees in pure mathematics from Warsaw University, Poland, in 1981 and 1985, respectively. He has worked at West Virginia University since 1989. Since 2006 he has held a position of adjunct professor in the Department of Radiology at the University of Pennsylvania. He is the author of three books and over 150 journal research articles. Ciesielski’s research interests include both pure mathematics (real analysis, topology, set theory) and applied mathematics (image processing, especially image segmentation). He is an editor of the Real Analysis Exchange, the Journal of Applied Analysis, and the Journal of Mathematical Imaging and Vision.