414/514 – Faber functions

Here is Shehzad Ahmed’s project from last term, on the Faber functions, a family of examples of continuous nowhere differentiable functions. Ahmed’s project centers on one of them, but the argument can be easily adapted to all the functions in the family.

As usual, the function is given as a series where the functions are continuous, and we can find bounds with and . By the Weierstrass -test, is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point a pair of sequences and with strictly decreasing to and strictly increasing to . The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function is differentiable at , then we have

In the case of the Faber functions, the functions add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points and ; moreover, the slopes accumulate at these peaks, and so the limit on the right hand side of the displayed equation above does not converge and, in fact, diverges to or .

Faber’s original paper, Einfaches Beispiel einer stetigen nirgends differentiierbaren Funktion, Jahresbericht der Deutschen Mathematiker-Vereinigung, (1892) 538-540, is nice to read as well. It is available through the wonderful GDZ, the Göttinger Digitalisierungszentrum.