2 1. The Real Line
ky
•^
x
Figure 1.1. A continuous and differentiable function over the rationals
If the reader thinks it is not the most general solution, then can she find
another solution?
After a little thought she may observe that if g(t) is a solution of (*)
and we set
/(*) = 9(t) ~ j
then f'(t) = 0 and the statement that (**) is the most general solution of
(*) reduces to the following theorem.
Theorem 1.1. (Constant value theorem.) Iff : R R is differentiable
and
ff(t)
0 for all t eR, then f is constant.
If this theorem is 'intuitively clear' over R it ought to be intuitively clear
over Q. The same remark applies to another 'intuitively clear' theorem.
Theorem 1.2. (The intermediate value theorem.) If f : R R is
continuous, b a and f(a) 0 f(b), then there exists a c with b c a.
such that f(c) = 0.
However, if we work over Q both putative theorems vanish in a puff of
smoke.
Example 1.3. / / / : Q Q is given by
f(x) = -l
zfx22,
f(x) = l otherwise,
then
(i) f is a continuous function with /(0) = 1, /(2) = 1, yet there does
not exist a c with /(c) = 0,
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