病的な関数

関数の不連続性について

Def. set of discontinuity
Given a function f:R→R, define Df⊂R
to be the set of points where the function f fails to be continuouts.

Def. Fσ set
A set that can be written as the countable union of closed sets
is in the class Fσ

Th. 
Given f:R→R be an arbitrary function.
Then, Df is an Fσ set.

Cor. 
(with Baire's category theorem, the set of irrational points is not Fσ set.)
There is no function f that is continuous at every rational point
and discontinuous at every irrational point.

Def. Dirichlet's function


a nowhere-continuous function on R

Def. Modified Dirichlet's function


not continuous at every point x≠0

Def. Thomae's function 1875


t(x) fails to be continuous at any rational point.
whereas t(x) is continuous at every irrational point on R.
それぞれQに収束する点列・Iに収束する点列をとってみれば分かる。

 は，区間(0,1]で連続かつ有界であるが、一様連続ではない

関数の微分可能性について

Def. Weierstrass 1872
a class of continuous nowhere-differentiable function

where the values of a and b are carefully chosen.

Def. Takagi

where 

Cor. of Baire's Category theorem
世の中の連続関数はだいたい Weierstrass 関数みたいに，全域で微分不能

Th. Lebesgue 1903
a continuous, monotone function would have to be differentiable at almost every point in its domain.