Is this function of bounded variation?

Question

Consider Riemann's function defined on $\mathbb{R}$,

$$ R(x) = \sum_{n=1}^\infty \frac{\sin n^2 x}{n^2} . $$

If you graph it, you can see that it shows a lot of zigzags.

Hence, the question is, is this function of bounded variation in $(0,2 \pi )$?

Answer 1

I am just sharing my opinion. Please don't take it as an answer,

Differentiate $R(x)$ and take absolute value we get, $$|R'(x)|\le\sum_{n=1}^\infty |\cos (n^2x)|$$

Is the RHS is bounded above by some quantity? If not then it is not of Bounded Variation.

Note: $|\cos x|\le 1,\forall x\in\mathbb{R}$

Answer 2

Remember that functions of bounded variation are differentiable almost everywhere. See here for example that your $R(x)$ is differentiable nowhere. Thus by contraposition, it is not of bounded variation.