5.4 Theorems about continuous and differentiable functions 313and is decreasing over the interval < x < 1, and why B4(x) has exactly two
zeros between 0 and 1. Continue this investigation until general conclusions
about the functions B (x) and their graphs have been reached.5.4 Theorems about continuous and differentiable functions
It is possible to look at Figure 5.42 and others more or less like it and claim
that these figures provide experimental evidence supporting the follow-
ing theorem of which we shall give a stronger version in Theorem 5.52.
Theorem 5.41 If L is a chord joining two pointson the graph of a
differentiable function, then there must be at leastone point on the graph at
which the tangent is parallel to L.
There are at least two reasons why this theorem is surprising. It is
thoroughly important, and it is impossible to prove it without makinguse
of some substantial mathematical machinery that has not yet appeared in
this course. The source of the difficulty
can be stated very crudely by saying that
Theorem 5.41 would be false if there were
"holes" in the set of real numbers so that
the graph of Figure 5.42 contains no points
having x coordinates x, and x2. To prove
Theorem 5.41, and for many other purposes,
we need a property or postulate or axiom a xi x2 b
which guarantees that the set of real numbers
is complete. While several different equiva-Figure 5.42lent axioms can be given, the following one involving a fundamental idea
of Dedekind (1831-1916) is in some respects the most natural one to
adopt.
Axiom 5.43 (Dedekind) Let the set of real numbers be partitioned
into two subsets .4 and B (see Figure 5.44) in such a way that (i) each reala
t
Figure 5.44number is put into either 14 or B, (ii) each of 4 and B contains at least one
real number, and (iii) if xl is in z1 and x2 is in B, then xl < xv Then
there is a real number (the partition number xi) which is either the
greatest number in A or the least number in B.
Once again we are in a position where we should know something about
our present state and prospects for future development. To attain full
comprehension of the Dedekind axiom, and the manner in which it is
used to prove basic theorems of mathematical analysis, is not a short
task. Experience shows that, except in unusual special circumstances,
it is quite unreasonable to suppose that enough time is available in a first