As is known to all, Dirichlet function is define: D(x)=0, when rational and D(x)=1, when rational. The upper limit of the Riemann sum goes to 1 and the lower limit goes to 0. In theory, can the Riemann sum be any value between 0 and 1, corresponding to various partitions?
2026-03-29 06:30:23.1774765823
possible values of the Riemann integral of Dirichlet function
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Yes, take a partition consisting of two subintervals, $[0,a]$ and $[a,1]$, and take in your Riemannsum the value at a rational point in the first interval and an irrational in the second. The value of the Riemann sum will be $$1\cdot(a-0)+0\cdot(1-a)=a.$$ This clearly works for any $a$ between $0$ and $1$.