How to prove that a function is Riemann integrable?

Question

I am given the function

$$ f : [−\pi/2, \pi/2] \rightarrow \mathbb{R} $$

where

$$ f(x) = \begin{cases} 1 & \text{if } \quad \sin(1/x) > 0, x \neq 0 \\ -1 & \text{if } \quad \sin(1/x) < 0, x \neq 0 \\ 0 & \text{if } \quad \sin(1/x) = 0, x \neq 0 \\ 0 & \text{if } \quad x = 0 \end{cases}$$

and I am asked if the function is Riemann integrable and why.

Answer 1

Here is a powerful theorem which you may keep in mind:

A function $f: [a,b] \to \mathbb{R}$ is Riemann-integrable on $[a,b]$ if and only if it is bounded and the measure of the set of discontinuities of $f$ has measure $0$.

So, with this in mind. Where is your function discontinuous?