Suppose $f: [-2,3] \longrightarrow \mathbb{R}$ is defined by $ f(x) = \left\{ \begin{array}{l l} 2|x| + 1, & \text{if $x$ is rational}, \\ 0, & \text{if $x$ is irrational}. \end{array} \right.$
Prove that $f$ is not Riemann integrable.
We know that the lower integral is $0$ and the upper integral is $18$, then because they are not equal $f$ is not Riemann integrable.
Is this correct? Thanks!