Darboux sums of the Dirichlet function

Definitions. For a partition $1=x_0<x_1<\dots<x_n=2$, let $m_i=\inf_{[x_{i-1},x_i]}f$ and $M_i=\sup_{[x_{i-1},x_i]}f$. Then
$$s_\sigma(f)=\sum m_i(x_i-x_{i-1}),\qquad S_\sigma(f)=\sum M_i(x_i-x_{i-1}).$$

Values. Each subinterval has positive length and so contains a rational (where $f=0$) and an irrational (where $f=1$). Hence $m_i=0$ and $M_i=1$ for all $i$, and
$$s_\sigma(f)=0,\qquad S_\sigma(f)=\sum(x_i-x_{i-1})=1.$$

Conclusion. $s(f)=0$ and $S(f)=1$. They differ, so $f$ is not Riemann integrable.

Remark. No refinement of the partition helps: the function oscillates fully on every tiny interval. This example motivated Lebesgue's theory, in which this function has integral $1$, since the rationals have measure zero.