352 Functions, graphs, and numbers
g(x) > f(x) for each x for which x > xo. Progress with the theory depends upon
the basic fact that if f and g are rational functions, then f(x) = g(x) for each x
or there is a number x0 such that f(x) < g(x) when x > x0 or there is a number
xo such that f(x) > g(x) when x > xo. This basic fact depends upon the fact
that if h is a rational function, then either h(x) = 0 for each x or there is a number
xo such that h(x) is continuousand positive or continuous and negative when
x > X. It follows that if f and g are rational functions, then one and only one
of the three relations f < g, f = g, f > g is valid. Thus the set of rational func-
tions is, like the set of real numbers, now an ordered field. Let fo, the zero function,
be the rational function for which fo(x) = 0 for each x. Our algebra of rational
functions is said to have the Archimedes property (or to be Archimedian) if to
each pair of functions f and g for which f > fo and g > fo there corresponds an
integer n for which of > g. This is of interest to us because we have proved,
with the aid of the Dedekind axiom, that the algebra of real numbers is Archi-
median. It could be presumed that the algebra of rational functions is so much
like the algebra of real numbers that the algebra of rational functions must be
Archimedian. However, the presumption is false, the algebra of rational func-
tions is not Archimedian. To prove this, let f and g be the rational functions
for which f(x) = x and g(x) = x2. Careful applications of our definitions then
imply that f > fo, g > fo, and of < g for each integer n. Thus our algebra of
rational functions is not Archimedian. When all matters which we have dis-
cussed are thoroughly understood, it becomes clear that the Archimedian prop-
erty of the algebra of real numbers is not a consequence of those properties of real
numbers that are ordinarily stated in elementary arithmetic and algebra. Alge-
bra books that give adequate treatments of matters relating to order relations,
bounds, limits, Dedekind partitions, and Archimedes properties are said to be
modern. We have seen some of the reasons why knowledge of modern algebra
is considered to be an essential part of a mathematical education.
10 While consideration of the matter is usually reserved for more advanced
courses, we have enough equipment to understand, and perhaps even prove,
basic facts involving functions of bounded variation. Let f(t) be defined over
a 5 t <= b and let a < x <= b. Supposing f such that T(x) exists (is finite) we
define numbers T(x), P(x), and N(x) by the formulas
(1)R
l.u.b. If(tk) - f(tk-1)I = T(x)
k=1(2) l.u.b. [ I f(tk) - f(tk-1)I + V(4) - f(tk-01 I = P(x)
k=1(3) l.u.b. 2 I If( ) - f(tk-1)I - [f(tk) - f(tk_1)I I = N(x).In each case, the least upper bound is the least upper bound of sums obtained for
partitions P of the interval a 5 t 5 x. The function f is said to have bounded
variation (the term finite variation would be better) over the interval a < t 5 b.
Let T(a) = P(a) = N(a) = 0. The numbers T(x), P(x), and N(x) are, respec-
tively, the total variation, the positive variation, and the negative variation off over