452 Chapter 7 • The Riemann Integral- An example^23 of a bounded open set A such that XA is not integrable on
any closed interval containing A:
Let 0 < E: < ~' {rn : n EN} denote the set of rational numbers in [O, 1],
Jn = ( r n - 2 0:,. , r n + 2 cn ) , and A = LJ:= 1 Jn. Let a = inf A and b = sup A.
By considering ~(XA, P) and S(xA, P) for partitions P of [a, b], show that
A and XA have the desired properties. - Show^23 that the set A of Exercise 6 can be written as the union of pairwise
disjoint open intervals, A= LJ:=l In (see Exercise 3.1.23). Then Vn E N,
let fn = X!i + XI 2 + · · · + Xn· Prove that Un} is a sequence of Riemann
integrable functions on I = [a, b] such that Vx E J, the sequence of
numbers Un(x)} converges, but the limit function f(x) = n->oo lim fn(x) is
not Riemann integrable on I. - Exercises 6 and 7 were suggested by an anonymous reviewer of an early version of the
manuscript.