1549901369-Elements_of_Real_Analysis__Denlinger_

398 Chapter 7 • The Riemann Integral

(c) If f is integrable on [a, b] and \:Ix E [a, b], lf(x)I ::::; M, then 11: fl <

M(b-a).

(d) If f and g are integrable on [a, b] and \:Ix E [a, b], f(x) < g(x), then

l: f::::; l: g.

Proof. Exercise 5. •

THE COMPOSITION THEOREM AND CONSEQUENCES

The composition of two continuous functions is continuous (Theorem 5 .1.14)
and the composition of two differentiable functions is differentiable (the chain
rule). We might expect that the composition of two integrable functions is al-
ways integrable. That this is not the case, however, is demonstrated by the
following example.

Example 7.5.3 Let T denote Thomae's function, defined in Example 5.1.12,
and g denote the characteristic function of the half-open interval (0, l]. Then
T, g: [O, l] -+ [O, l] are both integrable on [O, l], but go Tis not integrable on
[O, l].

Proof. Exercise 7. D

The following theorem shows that if f is integrable and g is continuous, the
composite function g o f is integrable. Its proof may seem a bit complicated,
and you may be tempted to give up on it. But the theorem is quite powerful,
yielding many significant results easily, such as the corollaries that follow.

Theorem 7.5.4 (Algebra of the Integral, III-The Composition The-

orem) Suppose f :[a, b] -+ JR is integrable on [a, b]. Further, suppose f[a, b] ~

[c, d] and g : [c, d] -+ JR is continuous. Then the composition go f : [a, b] -+ JR

is integrable on [a, b].

Proof. Suppose f:[a, b] -+JR is integrable on [a, b], f[a, b] ~ [c, d] = J, and

g : J-+ JR is continuous on J. Let c > 0. Since g is continuous on the compact

set J = [c, d], it is bounded there, so :3 A < B in JR 3

Vt E J, A ::::; g(t) ::::; B.

Also, g is uniformly continuous on J, by Theorem 5.4.7. Hence, :38 > 0

such that 8 < c, and

Vs, t E J, Is - ti < 8::::} lg(s) - g(t)I < c. (14)