1.2. Limits
3
(ii) f is a differentiable function with f'(x) = 0 for all x, yet f is not
constant.
[We sketch f in Figure 1.1.]
Sketch proof. We have not yet formally defined what continuity and dif-
ferentiability are to mean. However, if the reader believes that / is discon-
tinuous, she must find a point x G Q at which / is discontinuous. Similarly,
if she believes that / is not everywhere differentiable with derivative zero,
she must find a point x G Q for which this statement is false. The reader
will be invited to give a full proof in Exercise 1.16 after continuity has been
formally
defined2.
A
The question 'Is (**) the most general solution of (*)?' now takes on
a more urgent note. Of course, we work in R and not in Q, but we are
tempted to echo Acton ([1], end of Chapter 7).
This example is horrifying indeed. For if we have actually seen
one tiger, is not the jungle immediately filled with tigers, and
who knows where the next one lurks.
Here is another closely related tiger.
Exercise 1.4. Continuing with Example 1.3, set g(t) t + f(t) for all t.
Show that g'(t) = 1 0 for all t but that g(-8/5) g(-6/5).
Thus, if we work in Q, a function with strictly positive derivative need
not be increasing.
Any proof that there are no tigers in R must start by identifying the
difference between R and Q which makes calculus work on one even though
it fails on the other. Both are 'ordered fields', that is, both support opera-
tions of 'addition' and 'multiplication' together with a relation 'greater than'
('order') with the properties that we expect. I have listed the properties in
the appendix on page 357, but only to reassure the reader. We are not
interested in the properties of general ordered fields but only in that partic-
ular property (whatever it may be) which enables us to avoid the problems
outlined in Example 1.3 and so permits us to do analysis.
1.2. Limits
Many ways have been tried to make calculus rigorous and several have been
successful. In this book I have chosen the first and most widely used path,
via the notion of a limit. In theory, my account of this notion is complete
in itself. However, my treatment is unsuitable for beginners and I expect
2We
use to mark the end of a proof and to mark the end of an argument which we do
not claim to be a full proof.
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