446 Chapter 7 • The Riemann Integral8E:
Putting (46) and (47) together, 5 I:; L.i < -. Therefore,
iEN 2
c
I:: L.i < -. •
iEN 2Theorem 7.9.5 (Lebesgue's Criterion Is Necessary for Integrability)
If f is bounded and int egrable on [a, b], then the set of discontinuities off in
[a, b] has measure 0.Proof. Suppose f is bounded and integrable on [a, b]. Let E > 0. As noted
00
in Remark 7.9.3 (f), the set of discontinuities off in [a , b], is S = LJ S 1 ;n(J).
n=l
By Theorem 7.9.4, each S 1 ;nU) can be covered by a collection Cn of finitely
many open intervals; i.e., Cn = {In1,ln2, .. · ,lnkn} of total length less than
c-. Then
2n
00
(a) Scan be covered by the collection C = LJ Cn of all the intervals in all
n=l
of these collections. That is, S s;; U C = n^00 ~l (ikn ~l Ini ) ·
(b) C is a countable collection of open intervals, since it is the union of
countably many finite collections of open intervals.
( c) The total length of all the intervals in C is less than f: !__ = c.
n=l 2n
Therefore, S has measure zero. •Lemma 7.9.6 Suppose f :[a, b] ---+ JR is bounded, and c > 0. If '<Ix E [a, b],
w1(x) < E, then 3 5 > 0 3 V closed intervals I s;; [a, b] of length l(I) < 5,
WJ(I) < E:.
That is, if the oscillation off at every point of [a, b] is less than E, then
there is some 5 > 0 such that the oscillation of f is less than c on all closed
subintervals of [a, b] with length less than 5.
Proof. Suppose f : [a, b] ---+ JR is bounded, and 3 c > 0 3 Vx E [a, b],
w1(x) < E. Keep c fixed throughout the remainder of the proof. Now '<Ix E [a, b],
WJ(x) = 8_,Q+ lim WJ (N&(x) n [a,b]). Hence '<Ix E [a,b], 35 x > 0 3
(48)To simplify notation, '<Ix E [a, b], let Ux = N&x; 2 (x), the neighborhood of x
of radius ~bx.