Chapter 5.2, Problem 83E

A nonintegrable function Consider the function defined on [0, 1] such that f(x) − 1 if x is a rational number and f(x) = 0 if x is irrational. This function has an infinite number of discontinuities, and the integral ${\displaystyle {\int }_0^{1}f(x)dx}$ does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n. (Hint: Between any two real numbers lie a rational and an irrational number.)