1549901369-Elements_of_Real_Analysis__Denlinger_

7.4 Basic Existence and Additivity Theorems 387

Theorem 7.4.2 (Additivity of the Integral, I) If f is integrable on [a, b]

then l::/c E (a, b), f is integrable on [a, c] and [c, b], and l: f = l: f + J: f.

Proof. Suppose f is integrable on [a, b] and c E (a, b). Since f is integrable

on [a, b], l: f = l: f. By Lemma 7.4.1, this implies

l: f + l: f = l: f + l: f. Thus,

(13)

By Theorem 7.2.7, the left side of Equation (13) is :S 0, and the right side
is ?: 0. Thus, both sides must be 0, from which we conclude that

l:f = l:f and

Therefore, f is integrable on [a, c] and [c, b]. Finally, using Lemma 7.4.l
and the definition of integrability, we have

Corollary 7.4.3 If f is integrable on [a, b], and P = {xo, X1, · · · , Xn} is any
partition of [a, b], then f is integrable on [xo, x1J, [x1, x2], · · · , and [xn-1, xnJ,
and

Proof. Exercise 2. •

Corollary 7.4.4 If f is integrable on [a, b], then f is integrable on any closed
subinterval [c, d] ~ [a, b], where c < d.

Proof. Exercise 3. •

The next theorem looks like one we have already seen (Theorem 7.4.2), but
is more like a converse of it. You may have to look twice to see the difference.

Theorem 7.4.5 (Additivity of the Integral, II) If f is integrable on [a, c]

and on [c, b], where a < c < b, then f is integrable on [a, b], and l: f =

l: f + l: f.