Exercise 8.1.10

Subintervals with irrational tags do not contribute to $R(g,P)$, so the maximum value of $R(g,P)$ is the sum of the lengths of the subintervals with rational tags. This can be no more than ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{x\in Q}\delta (x)=𝜖$. To show strict inequality, note that any subinterval must contain more than one rational number, and so some of the subinterval lengths are “missing”.