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,