1549901369-Elements_of_Real_Analysis__Denlinger_

7.2 The Riemann Integral Defined 365

Proof. Suppose a < b. For every partition P of [a, b], and for every i =

1,2, · · · ,n, we have mi= 0, and Mi= 1, so

n n

9-(f, P) = L md:-,i = L 06.i = 0, and hence, l: f = 0. Similarly,

i=l i=l

n n n

S(f, P) = L Mi6i = L 16.i = L 6i = (b-a), and hence l: f = (b-a).

i=l i=l i=l

Thus, l: f "I-l: f , from which it follows that f is not integrable on [a, b].

Example 7.2.11 Consider the characteristic function^5 of a closed interval, say

{

f = X[l,3]> given by f(x) =^1 if^1 - < x < - 3, }. Prove that f is integrable on

0 otherwise

[O, 5] and find l: f.

y

2 3 4 5 x

Figure 7.3

Solution. Our intuitive understanding of the integral as area (see Figure

7.3) leads us to expect that l: f = 2, so we start with that expectation.
(a) Let P = {O, 1 , 3, 5}. Then Pis a partition of [O, 5], and

9-(f, P) = m161 + m262 + m363

= 0·1+1·2+0·2

= 2.

Thus, since l: f is the supremum of all the lower sums, l: f?: 9-(f, P) = 2.

5. The characteristic function of a set was defined in Exercise 5.2.5.