Can One Visualize a Continuous Nowhere Differentiable Function?

Abstract

When students first encounter an example of a continuous nowhere differentiable function in a real analysis course, what do they visualize? It is not easy to visualize an infinite sum of functions when the partial sums get increasingly complicated. We offer a geometric approach via the graph of such a function.

Our theme is based on the fact that, if the graph of a function intersects all non-vertical lines in a special way, then that function cannot have a derivative at any point. We will require that at each point P on the graph G of the function there must be many lines L through P having P as a limit point of the intersection $G\cap L$. This picture is easy to imagine but impossible to represent graphically: it avoids all of the computational complications that faced nineteenth-century mathematicians when they first attempted to describe these functions.

MSC:

• 26A27

Notes

¹ See [5] for the details. In countably many directions there will be a single line whose intersection with the graph has two isolated points.

Additional information

Notes on contributors

Andrew M. Bruckner

ANDREW M. BRUCKNER obtained his Ph.D. from UCLA in 1959 and joined the faculty at UCSB that year. He retired from UCSB in 1994 as Professor Emeritus. Through 1990–1994 he served as an editor for the AMS Proceedings. In 2013 he was elected as a Fellow of the American Mathematical Society. The Andy is an award given each year at the annual Symposium in Real Analysis, which is organized by the Real Analysis Exchange. [email protected]

Judith B. Bruckner

JUDITH B. BRUCKNER obtained her Ph.D. in mathematics from UCLA in 1960 with a dissertation in the field of Riemann Surfaces. She worked in industry in the areas of pattern recognition, computer design, and operating systems. She has taught at several colleges and collaborated in the publication of papers and books, mostly in real analysis. [email protected]

Brian S. Thomson

BRIAN S. THOMSON received his mathematical training at the University of Toronto and the University of Waterloo. In 1968 he joined the faculty of the newly-created Simon Fraser University on the west coast of Canada. He has served on the editorial boards of the Real Analysis Exchange and the Journal of Mathematical Analysis and Applications. In 2021 he was the recipient of the MAA Halmos-Ford Award for an article he published in this Monthly. [email protected]