Consider the function defined on [0, 1] such that f(x) = 1 if x is a rational

Question:

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 does not exist. Show that the right, left, and midpoint Riemann sums on regular partitions with n subintervals equal 1 for all n. Between any two real numbers lie a rational and an irrational number.