Continuous Nowhere Differentiable Functions

In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by $$ W(x) = \sum_{k=0}^{\infty} a^k\cos(b^k\pi x)\text{,} $$ where $a$ is a real number with $0 < a < 1$, $b$ is an odd integer and $ab > 1 + 3\pi/2$. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass’ nor Riemann’s function was the first such construction.

The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differentiable. Around 1860, the Swiss mathematician Charles Cell\’erier also discovered (independently) an example which unfortunately wasn’t published until 1890 (posthumously).

After the publication of the Weierstrass function, many other mathematicians made their own contributions. We take a closer look at many of these functions by giving a short historical perspective and proving some of their properties. We also consider the set of all continuous nowhere differentiable functions seen as a subset of the space of all real-valued continuous functions. Surprisingly enough, this set is even “large” (of the second category in the sense of Baire).

Author: Thim, Johan

Source: LuleƄ University of Technology

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