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
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
More precisely, what is the general solution
of the equation
(*) g'(t) = t2?
We know that
is a solution but, if we have been well taught, we know
that this is not the general solution since
(**) g(t) = j + c,
with c any constant, is also a solution. Is (**) the most general solution of
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.
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