1549901369-Elements_of_Real_Analysis__Denlinger_

7.9 *Lebesgue's Criterion for Riemann Integrability 449

Now, L (Mi - mi)6.i ::; L (M - m)6.i

iEN1 iEN1

= ( M - m) l: l (Ii)

iEN1

c

< (M - m) 2(M - m) - 2 (53)

[by (49)]

M

Note that, i E N2 :::} [xi-1,Xi] ~ n Ji by (50)

i=l

and WJ ([xi-1, xi]) < J by (51)

(54)

Putting together (52)- (54), we have S(f, P) - $_(!, P) < r::. Therefore, by
Riemann's condition for integrability (7.2.14) f is integrable on [a , b]. •

Examples 7.9.8

Lebesgue's criterion is extremely powerful. It can be applied to virtually ev-
ery bounded function we have encountered in this course to determine whether
that function is integrable on a given interval. In the following examples, re-
member that countable sets have measure 0.
(a) Bounded, piecewise continuous functions have only finitely many dis-
continuities, so they are integrable on every closed interval. Likewise, monotone
functions are integrable on every closed interval since they have at most count-
ably many discontinuities (see Theorem 5.2.20). The same conclusion carries
over to piecewise monotone functions.

(b) The function f ( x) = sin ( ~) is continuous except at x = 0, so it is
integrable on every closed interval. Similar conclusions hold for other functions
related to it.

(c) Thomae's function, defined in Example 5.1.12, is integrable on [O, 1]
because the set of its discontinuities in [O, 1] is the set of rational numbers in
[O, 1], which is a countable set. (See also Exercise 7.4.17.)

(d) The function f: [O, l]--> IR defined in Example 7.4.10 by

f ( x) = { 1 if x = 1 / n for some n E N, }

0 otherwise

is integrable on [O, l]. The set of its discontinuities in [O, 1] is { ~ : n E N}, which
is countable.