Is the function $f(x) = \sum_{n=1}^\infty \cos(nx)/\sqrt{n}$ differentiable on $(0,\pi)$?

Question

By term-by-term differentiation, I would expect this function is not differentiable almost everywhere. However, it is claimed that this function is monotonic, which would imply that it is differentiable almost everywhere. See the thread below:

Is $f(x) = \sum_{n\geq 1} \frac{\cos n x }{\sqrt{n}}$ monotonic on $(0,0.1)$?

So, which one is true?