Why do we bother

Why do we bother

Analysis consists in the difficult proofs

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

Theorem 1.

Theorem 1.1. (Constant value theorem.) /// : Ш. —> M. is differentiable and f'(t) = 0 for all teK, then f is constant.
If this theorem is 'intuitively clear' over Ш. it ought to be intuitively clear over Q. The same remark applies to another 'intuitively clear' theorem.

Sketch proof

We have not yet formally defined what continuity and differentiability are to mean. However, if the reader believes that / is discontinuous, she must find a point x 6 Q at which / is discontinuous.

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 #(-8/5) > #(-6/5).
Thus, if we work in Q, a function with strictly positive derivative need not be increasing.