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 $F=\sum_n f_n$ where the functions $f_n$ are continuous, and we can find bounds $M_n$ with $\|f_n\|\le M_n$ and $\sum_n M_n<+\infty$. By the Weierstrass $M$-test, $F$ is continuous. Also, the strategy for nowhere differentiability is typical: We associate with each point $x$ a pair of sequences $(a_n)_{n\ge0}$ and $(b_n)_{n\ge0}$ with $a_n$ strictly decreasing to $x$ and $b_n$ strictly increasing to $x$. The key lemma (shown, for example, in Johan Thim’s Master thesis available here) is that, if a continuous function $f$ is differentiable at $x$, then we have

$\displaystyle f'(x)=\lim_{n\to\infty}\frac{f(a_n)-f(b_n)}{a_n-b_n}.$

In the case of the Faber functions, the functions $f_n$ add `peaks’ in the neighborhood of any point, and the locations of these peaks can be used as the points $a_n$ and $b_n$; 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 $+\infty$ or $-\infty$.

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.