[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む49 (658ﾚｽ)
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80(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:16:04.01 ID:dCRrvhl7(16/27) AAS
>>79 つづき

http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/maths/real_analysis_2009_abstract.pdf
典型的連続関数のDini微分（著者最終稿）斎藤新悟 実解析学シンポジウム2009報告集，pp. 25-33.

（上記の関連参考：出典URL）
http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/maths/papers.html
斎藤新悟 出版物
http://www.artsci.kyushu-u.ac.jp/~ssaito/jpn/
斎藤新悟 九州大学基幹教育院准教授 1981年大阪府生まれ 東京大学理学部数学科卒業

つづく

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81(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/01(月) 17:16:33.36 ID:dCRrvhl7(17/27) AAS
>>80 つづき

（以前のスレから関連抜粋）
スレ46 2chｽﾚ:math
＜引用＞
http://www.unirioja.es/cu/jvarona/downloads/Differentiability-DA-Roth.pdf
DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009
This paper has been published in Gazette of the Australian Mathematical Society, Volume 36, Number 5, November 2009, pp. 353{361.
(抜粋)
So, in this paper we
are going to analyze the dierentiability of the real function
fν(x) =0 if x ∈ R ＼ Q,
　　or =1/q^ν if x = p/q ∈ Q, irreducible,
for various values of ν ∈ R.

Theorem 1. For ν > 2, the function fν is discontinuous (and consequently
not dierentiable) at the rationals, and continuous at the irrationals. With
respect the dierentiability, we have:
(a) For every irrational number x with bounded elements in its continued fraction expansion, fν is differentiable at x.
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.
Moreover, the sets of numbers that fulfill (a) and (b) are both of them uncountable.

Theorem 2. For ν > 2, let us denote
Cν = { x ∈ R : fν is continuous at x },
Dν = { x ∈ R : fν is dierentiable at x }.
Then, the Lebesgue measure of the sets R ＼ Cν and R ＼ Dν is 0, but the four sets Cν, R ＼ Cν, Dν, and R ＼ Dν are dense in R.
(引用終り)

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