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Continuous nowhere differentiable functions

Other mathematicians of the second half of the 19th century shared Hermite's opinion, fearing that similar investigations into the foundations of mathematics would lead to harmful results.

As late as 1920, Jasek is said to have created a "sensation" when he revealed Bolzano's example of a continuous function that is neither monotone in any interval nor has a finite derivative at the points of a certain everywhere dense set. It has been pointed out that Bolzano's function is actually nowhere differentiable, though he neither claimed nor proved this. Bolzano discovered/invented this function about 1830.

More than 30 years after Bolzano's result, the following example by Weierstrass of a continuous function which has a derivative at no point was published:

$\displaystyle f(x) = \sum_{n=1}^\infty b^n cos(a^n \pi x)$, where $0 < a < 1$, $b$ is positive odd integer, and $\displaystyle ab > 1+\frac{3}{2}\pi$

A turn of the century address to the American Mathematical Society summarized the situations ....

Primary sources

• duBois-Reymond, P. (1875) Journal fur die reine und angewandte Mathematik, 79 pp. 29-31.