Continuous Nowhere Differentiable Functions

Abstract: 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similar construction (which was unpublished) in his own lectures as early as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).