08 February 2016

Darboux's beautiful monster

Monsters are not known for their beauty, see Dr. Frankenstein's. Yet, we proudly present

!! Dr. Darboux's beautiful monster !!

To the analyst, a monster is a function that is everywhere continuous yet has no derivative at any point. You can read everything about them in this monograph from 2015.

The first properly tamed specimen was presented to the mathematical world (astonishing and horrifying many) by Weierstrass. It may have had geometrical predecessors, but Weierstrass's monster was a series of cosines, cunningly put together. Its technical details were of the kind one would expect of intricate functions with fractal graphs. But in 1879 Darboux came with a quite general approach, including several (well, at least two) elegant specimens. The most beautiful and simple is this one:
Looks like a charming little exercise in Fourier series, no? You'll find the monstrous truth in Darboux's original paper here (p. 198, 2), or in my own self-contained version here, less than two pages of elementary stuff. Incidentally, the proof shows that, if the function were differentiable (quod non), its derivative would be
Just differentiate termwise, what else!

Viewed from a distance, the graph of the function (courtesy of Geogebra) looks like this:

Actually, this is just the sum of the first four terms—adding more didn't change the view in any noticeable way. The graph looks piecewise smooth, but this is just an optical illusion. If you were given the real graph (and not some rough approximation), you could zoom in indefinitely without ever discovering any smooth portion however small.