Calculus: Analytic Geometry and Calculus, with Vectors

344 Functions, graphs, and numbers

0 < x < 1. It is now time to seek the moral of this story. If we are not sure
whether the set of numbers we use in our analytic geometry and calculus is the
complete set of real numbers for which the Dedekind postulate is valid, then we
cannot be sure about the validity of the ideas that we need to enable us to do
our chores. It is, therefore, not enough to know the axioms usually given in
one way or another in elementary arithmetic and algebra and "finite mathe-
matics." We need, in addition, the Dedekind axiom or an equivalent axiom
which guarantees that we are using the complete class of real numbers in our
work. We now come to the problem. Tell whether it is necessary to use the
Dedekind axiom (or, what amounts to the same thing, to use consequences of the
Dedekind axiom or an equivalent axiom) in order to (a) prove the Rolle theorem
5.51, (b) prove the intermediate-value theorem 5.48, (c) define the area of a
rectangle to be the product of its ddimensions, (d) define the derivative of a given
function f, (e) prove existence of

r12

(1/x) dx.

5.7 Darboux sums and Riemann integrals This section can be

omitted from this course without damaging understanding of the rest of

the book. There can, however, be no doubt that students with serious

interest in pure mathematics should master it and that everyone else

should read it. The section gives substantial information about a stand-

ard way of attacking matters relating to existence of Riemann integrals.

Let f be defined and bounded over an interval a 5 x < b so that, for some

constants in and M, we have

(5.71) m < f(x) < M (ax5b).

As in our definition of Riemann sums, let P be a partition such as the one

shown in Figure 5.711 and, for each k, let xk be selected such that xk_1 <

X1* x2 x3 k x

0 1 0-;-0 -cam;-o -o-------------o

a=xp x1 x2 xk_1 xk x,,_1

Figure 5.711

xk S xk. Let Axk = xk - xk_I. For each k = 1, 2, , n,let

(5.712) mk = g.l.b. f(x), Mk = l.u.b. f(x)

xk_I 52 ;9xk xk_1 SX SXk

so that ink and Mk are respectively the greatest lower bound and the least

upper bound of f over the interval xk_1 S x =< xk. The numbers UDS(P)

and LDS(P) defined by

n n

(5.72) LDS(P) _ Mk dxk, UDS(P) _ Mk Axk

k1 k-1

are called the lower and upper Darboux (1842-1917) sums determined by