example of a non Riemann integrable function

Let $[a,b]$ be any closed interval andconsider the Dirichlet’s function $f:[a,b]\to ℝ$

Then $f$ is not Riemann integrable. In fact given any interval $[x_1,x_2]\subset [a,b]$ with one has

$$\underset{[x_1,x_2]}{sup}f(x)=1,\underset{[x_1,x_2]}{inf}f(x)=0$$

because every interval contains both rational and irrational points.So all upper Riemann sums are equal to $1$ and all lower Riemann sums are equal to $0$.