A topological space is a Baire space if a countable intersection of open dense sets is still dense in . Then Baire category theorem consists in:
Theorem: (Baire) A complete metric space is a Baire space.
It is a surprising consequence of Baire category theorem that almost all continuous functions are nowhere differentiable in the following sense: Let be the set of continuous functions endowed with the sup norm ; then
Property: The set of continuous nowhere differentiable functions is dense in .
Proof. For all , let
For convenience, let be the complement of in . We first show that is open, or equivalently that is closed.
Let be a sequence in converging to some . So for all , there exists such that for all , . By compactness, there is a subsequence converging to some . Let ; for large enough, and for all . Then, for :
When , we get . Therefore, hence is closed.
Then, we show that is dense in . Let and . Because is uniformly continuous (according to Heine theorem), there exists such that for all , implies . Now let such that for .
Introduce piecewise linear such that , and for all and . In particular, . Let and such that . Then
hence . Therefore, is dence in .
Because is a Banach space, is dense in according to Baire theorem. However, any fonction of is nowhere differentiable, consequently the set of nowhere differentiable functions is itself dense in .
Corollary: The set of continuous nowhere locally monotonic functions is dense in .
A monotonic function being almost everywhere differentiable, a nowhere differentiable function is nowhere locally monotonic and the assertion follows from the property. However, we can give a direct proof using Baire category theorem:
Sketch of proof. For with , let . If and , then there exists a piecewise linear function oscillating around such that and . Therefore, has empty interior. Moreover, is closed: let be a sequence in converging to some ; then, for all , , so .
The same argument can be used for .
According to Baire category theorem, the set of continuous functions locally monotonic at some point has empty interior.