1549901369-Elements_of_Real_Analysis__Denlinger_

370 Chapter 7 • The Riemann Integral

n

Then, S(f, P) - S..(f, P) = L(Mi - mi)6i

i=l n

= L (f(x~') - l(xD) 6.i

i=l n

< :Lc-6.i

i=l n

= EL 6.i = c-(b - a).

i=l

Hence, 3 partition P of [a, b] 3 S(f, P) -S..(f, P) < (b-a)c-. By Riemann's

criterion for integrability, l is integrable over [a, b]. •

EXERCISE SET 7.2

1. Prove Lemma 7.2.2. [Hint: compare the formulas for S..(f, P) and S(f, P).]


2. Justify the assertion made in the proof of Theorem 7.2.4: "It is sufficient
   to consider the case when Q contains exactly one point not contained in
   P."


3. Suppose a< band f:[a,b]--> R. Prove that if\ix E [a,b], m:::; l(x):::; M,

then m(b-a):::; l:l:::; l:l:::; M(b-a) and l:l-l:l:::; (M -m)(b-a).

4. Consider the characteristic function of an open interval, say l = xc 3 , 6 ),

{

defined by l(x) = lif3<x<6' }. Prove that l .. is mtegrable on [O, 10]

0 otherwise

and find (^10)
10
f. (See Example 7.2.11, but b eware: the interval (3, 6) is open
in this case.)

5. Consider the characteristic function of a single-point set, say l = X{s},

defined by l(x) = {

1

if x =

5

' } · Prove that l is integrable on [2, 9]

0 otherwise

and find l; f. (See Example 7.2. 11 and Exercise 4.)

6. For the function l(x) = 5 - 3x, on the interval [O, 2], use each given
   partition to find S..(f, P) and S(f, P): [Caution: This function is decreasing
   on the given interval.]

(a) P = {o,~,1,~,2} (b) P = {o,~,~,1,~,~,2}

(d) p = {O 'n' 1. i n' n' .2.... ' 2n} n