1549901369-Elements_of_Real_Analysis__Denlinger_

7 .5 Algebraic Properties of the Integral 399

By Riemann's condition for integrability (7.2.14) there is a partition 'P =
{ Xo, X1, · · · , Xn} of [a, b] such that

S(f, P) -$_(!, P) < o^2 •

For each i = 1, 2, · · · , n, we use the notation

mi(f) = inf{f(x): Xi- 1 ::; x::; xi} and

mi (g o J) = inf { (g o J) ( x ) : Xi-l ::; x ::; xi}

(15)

and similarly for Mi(f) and Mi(g of). Divide the set N = {1, 2, · · · , n} into
two subsets:

(a) Suppose i E N 1. Then

x,y E [xi-1,xi] =? lf(x) - f(y)I::; Mi(f) -mi(f) < o

=? l(g o J)(x) - ( 9 o J)(y)I < c by (14).

Thus, I: (Mi(g of) -mi(9 o J)) 6i ::; L c 6i = c L 6i

iEN1 iEN1 iEN1

::; c(b - a).

(b) Suppose i E N2· Then Mi(g o J) -mi(9 o J)::; B - A, so

M·(f) -m·(f)

Now i E N2 =? Mi(f) -mi(f) ~ o, so '

0

' ~ 1, so

I: 6i ::; I: Mi(f) -mi(f) 6i = ~ L [Mi(f) - m i(f)]6i

iEN2 iEN2 O O iEN 2

1 1

::; J [S(f, P) -$_(!, P)] < J · 82 = o by (15)

< c by definition of o.

Putting (17) and (18) together, we have

L (Mi(g o J) -mi(9 o f)) 6i < (B - A)c.

iEN2

(16)

(18)

(19)