At how many points in the open interval $(0, 2\pi)$ is $f(x) = [\sin x]$ not continuous?

Explanation

1. 01

   $[\sin x]$ can change only where $\sin x$ is an integer: at $x = \frac{\pi}{2}, \pi, \frac{3\pi}{2}$.

2. 02

   At $\frac{\pi}{2}$, $f$ is 0 on both sides but $f = 1$ at the point; at $\pi$ it drops from 0 to $-1$; at $\frac{3\pi}{2}$ it is $-1$ on both sides and at the point. So 2 points.