Why this function can't be used in Fourier expansion?

$$y = \arccos(\sin(2x))$$

I can't see why it can't be used in Fourier expansion series. It seems to me that it satisfies all the Dirichlet properties:

Periodic ? Yes, $\pi $.

Continuous ? Yes

Finite number of max/mín in a period? Yes, there is one maximum and zero mínimum.

Module of the integral converges? Yes,$\frac{\pi^2}{2}$

So, what is the problem with the function? Why can't it be used for Fourier expansion?

Solution 1:

The book is available here.

This is Exercise 12.3 on page 429 (12.9 Exercises). However in the book the function is written slightly differently:

(d) $\cos^{-1}(\sin 2x)$, $-\infty < x < \infty$;

Note that the author wrote $\cos^{-1}$ instead of $\arccos$.

The answer is given on page 431 (12.10 Hints and answers).

According to the answer, the "Dirichlet condition" for which it fails is the following:

(ii) it must be single-valued and continuous, except possibly at a finite number of finite discontinuities;

Thus I guess what the author has in mind is that the "function" $\cos^{-1}$ is not single-valued.