Is Dirichlet function Riemann integrable? "Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$. 
On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded and continuous almost everywhere, which the Dirichlet function satisfies.
On the other hand, the upper integral of Dirichlet function is $b-a$, while the lower integral is $0$. They don't match, so that the function is not Riemann integrable.
I feel confused about which explanation I should choose... 

The Dirichlet function $f$ isn't continuous anywhere. For every irrational number $x$, there is a sequence of rational numbers $\{r_n\}$ that converges to it. We have:
$$
\lim_{n\to\infty} f(r_n) = 1 \ne 0 = f(x)
$$
Thus, $f$ isn't continuous at irrational numbers. Rational numbers can be handled similarly.
