Bolzano-Weierstrass functions

Our rapid overview of mathematical monsters would not be complete without a brief reference to an interesting family of continuous, nowhere differentiable functions. They are occasionally referred to as Weierstrass-like but, for historical reasons, it seems more appropriate to call them Bolzano-Weierstrass-like . 

We consider a few nowhere differentiable functions. Some of them have such amazing properties that they have been given the names of the great mathematicians who invented them Bolzano, Cantor, Peano, Weierstrass, Koch, Van der Waerden, Sierpinski, and others. 

The manuscript of Bolzano s was discovered only in 1920. So that the example of a nowhere differentiable function found by Weierstrass in 1871 was deemed the first example of such a function. Nowadays many examples of nowhere differentiable functions exist. Let us consider a nowhere differentiable Bolzano function paying tribute to Bolzano as the first scientist who formulated a nowhere differentiable function (Fig. 1). 

Indeed, Bernard Bolzano [16] appears to have been the first (in a manuscript written around 1830 but published only in 1930) to provide an example of a continuous, nowhere differentiable function. A few years later, in 1872, Karl Weierstrass [17] showed that the function... 

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