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9.5 *A Glimpse Beyond the Horizon 581

Theorem 9.5.3 Every complete normed vector space is of second category.

The reason for including this theorem here is the following amazing discov-

ery:

Theorem 9.5.4 The set of all functions in C[a, b] that are differentiable some-

where in [a, b] is of first category in C [a, b].

By definition, the union of two first category sets must be a first category

set. Also, C[a, b] with the sup norm is a complete normed vector space, and so

must be of second category. These statements cannot both be true unless the

set of all everywhere continuous, nowhere differentiable functions on [a, b] is of

second category C[a, b]. This leads us to two remarkable conclusions:

(1) We have a proof of the existence of continuous, nowhere differentiable

functions on [a, b] that is valid without ever producing a single example.

(2) Among all functions that are continuous everywhere on [a, b], those
that are differentiable somewhere form a much smaller set than those that are
differentiable nowhere. (First category sets are much "smaller" than second
category sets.) This is reminiscent of the situation in the real number system:
The rational (nice) numbers are far outnumbered by the irrational (not so nice)
numbers, since the former form a countable set while the latter form an un-
countable set. Similarly, the algebraic numbers form a countable set and so are
far outnumbered by the transcendental numbers, which form an uncountable
set.

With these rather unsettling results,^17 we take our leave. May you enjoy
further study in real analysis. The subject is rich with treasures to discover!

17. For more details consult pages 63 - 65 and 70-72 of [16]