1549901369-Elements_of_Real_Analysis__Denlinger_

388 Chapter 7 • The Riemann Integral

Proof. Suppose f is integrable on [a, c] and on [c, b], where a < c < b.
Then f is bounded on [a, b] and

l:f = l:f + l:f by Lemma 7.4.l

= l: f + l: f since f is integrable on [a, c] and [c, b]

by Lemma 7.4.l

Therefore, f is integrable on [a, b]. The desired equation follows immedi-
ately from Theorem 7.4.2. •

Corollary 7 .4.6 If P = {xo,X1, · · · ,xn} is any partition of [a,b], and f is

integrable on each subinterval [xi-I, xi] created by this partition, then f is in-

tegrable on [a, b] and

l:f= f: ff.

i=l Xi-1

Proof. Exercise 5. •

The following theorem may seem rather innocuous, but it has amazing
consequences, as we shall see.

Theorem 7.4. 7 Suppose f : [a, b] --+ JR is bounded, and is integrable on every
proper^8 closed subinterval of the open interval (a, b). Then

(a) f is integrable on [a, b], and

(b) l: f = h->O+ lim t+h a f = h->O+ lim t-h a f = h->O+ lim t+-~ a f.

Proof. Suppose f : [a, b] --+ JR is bounded, and is integrable on every
closed subinterval of (a, b).

(a) Let 0 < E: < (b - a)/2. Then 0 < 2i:: < b - a, so a+ i:: < b - i:: and

rb f = ra+e:f + rb-e:f + Tf and rbf = ra+e:f + rb-e:f + rb f.

Ja Ja Ja+e: Jb-e: :!_!!_ _Ja_ Ja+e: Jb-e:

8. A proper interval is bounded and contains more than one point. See Definition 5.7.13.