Calculus: Analytic Geometry and Calculus, with Vectors

5.7 Darboux sums and Riemann integrals 347

All this gives the following theorem which involves the numbers L and

U defined in the sentence containing (5.73).

Theorem 5.75 If f is defined and boundedover a < x 5 b, then f is

Riemann integrable over a x _< b if and only if L = U. Moreover,

(5.744) holds when L = U = I.

This theorem and (5.731) imply the following useful theorem.

Theorem 5.751 4 function f is Riemann integrableover a <- x <_ b if

and only if to each e > 0 there corresponds a partition P such that

(5.752) UDS(P) - LDS(P) 5 e.

The above story provides ideas and results that are used in proofs of

the fundamental theorem (Theorem 4.26) on existence of Riemann

integrals. We shall use Theorem 5.751 to prove some less pretentious

theorems.

Theorem 5.76 If f is defined and monotone increasing (or monotone

decreasing) over a 5 x < b, then the Riemann integral

fa'f(x)
dx exists.
Let e > 0. Suppose first that f is monotone increasing so that Ax') -<
f(x") when a < x' < x" < b. Let P be a partition of a <-- x < b with
partition points xo, x1, , x as in Figure 5.711. Then

(5.761) UDS(P) - LDS(P) l.u.b. f(x) - g.l.b. Ax)] tlxk

k-1 Xk_L SZ Sxk zk_i <Z <Xk

n

_ I [f(xk) - f(xk-1)] OxkC [f(xk) - f(x1---1)]IPI

k=1 k=1

[f(b) - f(a)]IPI < e

provided IPI is sufficiently small. This and Theorem 5.751 establish

the result for the case in which f is monotone increasing. In case f

is monotone decreasing, the proof is exactly the same except that [f(xk) -

f(xk_1)] is replaced by [f(xk_1) - f(xk)] and [f(b) - f(a)] is replaced by

[f(a) - f(b)]-

It is easy to extend Theorem 5.76 to obtain a better theorem. A func-

tion f is said to be bounded and piecewise monotone over the closed

interval a S x 5 b if there is a constant M for which I f (x) I< M when

a < x S b and if there is a partition P of the interval a < x <- b such

that, whenever xk_1 and xk are two consecutive partition points, f is

monotone (maybe monotone increasing, maybe monotone decreasing)

over the open interval xk_1 < x < xk.

Theorem 5.762 If f is bounded and piecewise monotone over a < x < b,

then the Riemann integral fab f(x) dx exists.t

t It is sometimes said that this theorem is a poor-man's version of a stronger theorem

which says that f is integrable over a 5 x 5 b if f has bounded variation over a S x <_ b.

Problem 10 at the end of this section provides opportunities to rise above poverty.