Calculus: Analytic Geometry and Calculus, with Vectors

346 Functions, graphs, and numbers

denote, respectively, the least upper bound of all lower Darboux sums
and the greatest lower bound of all upper Darboux sums. These num-
bers are, respectively, the lower and upper Darboux integrals of f over the
interval a < x < b. It follows from (5.73) that, for each partition P2,
L < UDS(P2), and it follows in turn from this that L S U. Thus

(5.731) LDS(P) : L < U < UDS(P)

for each partition P.

There are bounded functions f for which L < U. For example, let

a = 0, let b = 1, and let f(x) = 0 when x is irrational and f(x) = 1 when

x is rational. The LDS(P) = 0 and UDS(P) = 1 for each P and there-

fore L = 0 and U = 1.

It can be proved that LDS(P) is nearL and UDS(P) is near U whenever

IPI (the norm of P) is small. This result, which is sometimes called the

Darboux theorem, means that to each e > 0 there corresponds a S > 0

such that

(5.74) ILDS(P) - Lj < e, IUDS(P) - UI < e

whenever IP! < S. This and (5.731) imply that, when JP1 < S, the

numbers LDS(P) and UDS(P) are respectively located in the left and

right intervals of Figure 5.741 when L < U and of Figure 5.742 when

1LDS(P) 1UDS(P) fLDS(P) `-UDS(P)

L-e L U U+! 1-e I I'k'e

Figure 5.741 Figure 5.742

L = U = I. Consider first the case in which L < U. Since each

Darboux sum can be approximated as closely as we please by a Riemann

sum having the same partition points, it follows that there exist Riemann

sums with norm JPJ < & which differ from L by less than e and that there

also exist Riemann sums with norm JPJ < S which differ from U by less

than e. It follows that if L < U, then f cannot be Riemann integrable

over a < x < b.

Consider next the case in which L = U = I. In this case

(5.743) I - e < LDS(P) 5 RS(P) < UDS(P) < I + e

whenever RS(P) is a Riemann sum formed for a partition P for which

JPJ < S. Therefore,

(5.744) f' f(x) dx = I,

the integral being a Riemann integral.