The following Fourier Series is continuous (by M-test) $$ \sum_{n=1}^{\infty}{\frac{\sin{n^n x}}{n^n}}$$
Is it differentiable anywhere, and if so where?
http://kryakin.org/at/hardy_1916_W.pdf covers similar things, but I didn't really understand it.
2026-04-24 21:27:40.1777066060
Differentiability of Fourier Series
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This is the sort of thing I knew something about years ago. I'm not sure of the answer to your question, but I can say that the series you give is right on the borderline.
Consider $f=\sum a_n\sin(n^n t)$. If $n^n a_n\to0$ (and the $a_n$ are real) then $f$ is in the little-oh Zygmund class, and hence is differentiable on a dense set. On the other hand if $n^na_n\to\infty$ then it's not hard to show that $f$ is nowhere differentiable. Your function is just exactly between those two classes. Possibly I'm being stupid again, or possibly it's not that easy to determine whether it's differentiable at at least one point.
Edit Rusty. Thinking about it a little more, I believe the function is nowhere differentiable; proving it would take a little space. What's unclear is whether it satisfies a $Lip_1$ condition at some point.