Chapter 1

The Real Line

1.1. Wh y do we bother?

It is surprising how many people think that analysis consists in the difficult

proofs of obvious theorems. All we need know, they say, is what a limit is,

the definition of continuity and the definition of the derivative. All the rest

is 'intuitively

clear'1.

If pressed they will agree that the definition of continuity and the defini-

tion of the derivative apply as much to the rationals Q as to the real numbers

R. If you disagree, take your favorite definitions and examine them to see

where they require us to use E rather than Q. Let us examine the workings

of our 'clear intuition' in a particular case.

What is the integral of

t2?

More precisely, what is the general solution

of the equation

(*) g'(t) = t2?

We know that

t3/3

is a solution but, if we have been well taught, we know

that this is not the general solution since

t3

(**) g(t) = j + c,

with c any constant, is also a solution. Is (**) the most general solution of

w?

If the reader thinks it is the most general solution then she should ask

herself why she thinks it is. Who told her and how did they explain it?

1 A good example of this view is given in the book [9]. The author cannot understand the

problems involved in proving results like the intermediate value theorem and has written his book

to share his lack of understanding with a wider audience.

l

http://dx.doi.org/10.1090/gsm/062/01 http://dx.doi.org/10.1090/gsm/062/01