Riemann's Almost Nowhere Differentiable Function
In the late nineteenth century, Riemann proposed the function as an example of a continuous function that is almost nowhere differentiable. This Demonstration shows the graphs of generalized versions of Riemann's function, defined by , where and are real parameters satisfying and , and corresponding functions are obtained by replacing by or .
Contributed by: Saurav Chittal, Malachi Robinson, Manisha Garg and A. J. Hildebrand.
(Based on an undergraduate research project at the Illinois Geometry Lab in Fall 2022.) (June 13)
Open content licensed under CC BY-NC-SA
Given two real parameters and , the generalized complex Riemann function is defined as
The generalized Riemann sine and cosine functions are defined as the imaginary and real parts of , respectively:
All three types of Riemann functions are known to be continuous but almost nowhere differentiable, and their graphs exhibit fractal behavior [1–3].
This Demonstration shows the graphs of the real-valued Riemann sine and cosine functions, and , over the interval , and the graph of the complex Riemann function, , , represented by a path in the complex plane. In the case when is an integer, the function is periodic with period 1, so the corresponding path is closed.
 F. Chamizo and A. Córdoba, "Differentiability and Dimension of Some Fractal Fourier Series," Advances in Mathematics, 142(2), 1999 pp. 335–354. doi:10.1006/aima.1998.1792.
 J. J. Duistermaat, "Selfsimilarity of 'Riemann's Nondifferentiable Function'," Nieuw Archief voor Wiskunde, 9(3), 1991 pp. 303–337. https://dspace.library.uu.nl/handle/1874/2601.
 J. Gerver, "More on the Differentiability of the Riemann Function," American Journal of Mathematics, 93(1), 1971 pp. 33–41. doi:10.2307/2373445.