1549901369-Elements_of_Real_Analysis__Denlinger_

368 Chapter 7 • The Riemann Integral

1

Thus, S.(f, Pn) ->

3

as n-> oo.

(c) Therefore, by (a), (b), and Theo rem 7.2.12 (c), f is integrable over [O, l]

and f 0

1

f = ~. D

In general it is quite difficult to show that a function is integrable without

developing some more powerful tools to use toward that end. The next theorem

is one such tool.

Theorem 7.2.14 (Riemann's Criterion for Integrability) A boundedfunc-

tion f :[a, b] -> JR is integrable on [a, b] if and only if

I \:/c; > 0, :J partition P of [a, b] 3 S(f, P) - S.(f, P) < c;. ,

[Equivalently, there is some positive constant K such that \:/c; > 0, :J partition

P of [a, b] 3 S(f, P) - S.(f, P) <Kc;.]

Proof. Exercise 16. •

The following theorem, which is equivalent to Riemann's condition, is oc-

casionally useful.

Theorem 7.2.15 (Equivalent Form of Riemann's Condition) A bounded

function f :[a, b] ->JR is integrable over [a, b] {::} there is one and only one num-

ber I such that\:/ partitions P of [a, b], S.(f, P) :::; I :::; S(f, P). (In this case, I = J: J.)

Proof. Exercise 18. •

The next two theorems demonstrate the power of Riemann's criterion in
showing that a function is integrable. These results are quite significant.

Theorem 7.2.16 If f is monotone on [a, b], then f is integrable on
[a,b].

Proof. Suppose f is monotone on [a, b].
Case 1 (f is monotone increasing on [a, b]): Let c; > 0. Then \:/x E [a, b],
f(a):::; f(x):::; f(b), so f is bounded on [a, b]. By the Archimedean property, :J
1 b

1. 1
   natura num er n > - ; 1.e., - < c;.
   c; n
   Consider the partition P = { x 0 , x 1 , x 2 , · · · , Xn} of n equally spaced points;
   b-a
   i.e., 6.i = --. Since f is monotone increasing, mi = f(xi-i) and Mi= f(xi) ·
   n