「病的な関数」の編集履歴（バックアップ）一覧はこちら
病的な関数 - (2010/11/10 (水) 11:30:10) の１つ前との変更点
追加された行は緑色になります。
削除された行は赤色になります。
== 関数の不連続性について ==
'''Def. set of discontinuity'''
Given a function f:R→R, define D<sub>f</sub>⊂R
to be the set of points where the function f fails to be continuouts.
'''Def. F<sub>σ</sub> set'''
A set that can be written as the countable union of closed sets
is in the class F<sub>σ</sub>
'''Th. '''
Given f:R→R be an arbitrary function.
Then, D<sub>f</sub> is an F<sub>σ</sub> set.
'''Cor. '''
(with Baire's category theorem, the set of irrational points is not F<sub>σ</sub> set.)
There is no function f that is continuous at every rational point
and discontinuous at every irrational point.
<math>\neg {}^\exists f:\mathbb{R} \to \mathbb{R} \mbox{ s.t. } D_f = \mathbb{I}</math>
'''Def. Dirichlet's function'''
<math>g(x) := \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}</math>
a nowhere-continuous function on R
<math>D_g = \mathbb{R}</math>
'''Def. Modified Dirichlet's function'''
<math>h(x) := \begin{cases} x & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}</math>
not continuous at every point x≠0
<math>D_h = \mathbb{R} \setminus \{ 0 \}</math>
'''Def. Thomae's function 1875'''
<math>t(x) := \begin{cases} 1 & x=0 \\ \frac{1}{n} & x=\frac{m}{n} \in \mathbb{Q}^\times \mbox{ where }m,n \mbox{ is coprime and } n>0 \\ 0 & x \notin \mathbb{Q} \end{cases} </math>
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。
<math>D_t = \mathbb{Q}</math>
== 関数の微分可能性について ==
'''Def. Weierstrass 1872'''
a class of continuous nowhere-differentiable function
<math>f(x) := \sum_{n=0}^\infty a^n \cos (b^n x)</math>
where the values of a and b are carefully chosen.
'''Def. Takagi'''
<math>\sum_{n=0}^\infty \frac{1}{2^n} h(2^n x)</math>
where <math>h(x) := |x| \, \mod [-1,1]</math>
'''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.
== 関数の不連続性について ==
'''Def. set of discontinuity'''
Given a function f:R→R, define D<sub>f</sub>⊂R
to be the set of points where the function f fails to be continuouts.
'''Def. F<sub>σ</sub> set'''
A set that can be written as the countable union of closed sets
is in the class F<sub>σ</sub>
'''Th. '''
Given f:R→R be an arbitrary function.
Then, D<sub>f</sub> is an F<sub>σ</sub> set.
'''Cor. '''
(with Baire's category theorem, the set of irrational points is not F<sub>σ</sub> set.)
There is no function f that is continuous at every rational point
and discontinuous at every irrational point.
<math>\neg {}^\exists f:\mathbb{R} \to \mathbb{R} \mbox{ s.t. } D_f = \mathbb{I}</math>
'''Def. Dirichlet's function'''
<math>g(x) := \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}</math>
a nowhere-continuous function on R
<math>D_g = \mathbb{R}</math>
'''Def. Modified Dirichlet's function'''
<math>h(x) := \begin{cases} x & x \in \mathbb{Q} \\ 0 & x \notin \mathbb{Q}\end{cases}</math>
not continuous at every point x≠0
<math>D_h = \mathbb{R} \setminus \{ 0 \}</math>
'''Def. Thomae's function 1875'''
<math>t(x) := \begin{cases} 1 & x=0 \\ \frac{1}{n} & x=\frac{m}{n} \in \mathbb{Q}^\times \mbox{ where }m,n \mbox{ is coprime and } n>0 \\ 0 & x \notin \mathbb{Q} \end{cases} </math>
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。
<math>D_t = \mathbb{Q}</math>
<math>\sin \frac{1}{x}</math> 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
<math>f(x) := \sum_{n=0}^\infty a^n \cos (b^n x)</math>
where the values of a and b are carefully chosen.
'''Def. Takagi'''
<math>\sum_{n=0}^\infty \frac{1}{2^n} h(2^n x)</math>
where <math>h(x) := |x| \, \mod [-1,1]</math>
'''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.
