A Simple Example of a Function, which is Everywhere Continuous and Nowhere Differentiable

This selection contains a simple example of continuous nowhere-differentiable function. An example quite similar to the one given here goes back to Bernard Bolzano [9]. Bolzano’s manuscript was written aboutl830 (well before the Weierstrass example),but published only a hundred years later (see [15], [10], pp. 30–32). See [19] for an interesting account of Bolzano’s mathematical work— it was remarkably modern in outlook, but mostly unpublished, and therefore not influential. Bolzano’s example is a self-affine function, slightly different from Kiesswetter’s example. Bolzano proved only that his function is non-differentiable on a dense set; but in fact it, too, is nowhere differentiable.