It is known that the Riemann function
$$ \sum_{n\geq 1 } \frac{\sin n^2 x }{n^2} $$
is differentiable only at countably many points.
How about the functions
$$\sum_{n\geq 1 } \frac{\sin n^2 x}{n^{5/2}}, \quad\;\; \sum_{n\geq 1 } \frac{\cos n^2 x}{n^{5/2}} ? $$
I plotted their graphs by taking the first 100000 terms. It is not so suggestive.
The problem can of course be generalized to other exponents.
