ピエロのアップしたＰＤＦ（下記）に証明があるが、下記無理数を(a)連分数展開可能な無理数の点と、(b)そうでない無理数で微分出来ない点に分け、 (a)は微分可能で、”(a) and (b) are both of them un-countable.”だと。まあ、これは私の手では独力では証明できないと悟った
事実、筆者もP2 "Actually, a big part of this study has already been done in the literature; see, for instance, [2, 3, 6, 7]. Here we present some results that are already known (usually whith a dierent proof), and some that seem to be new."とあって、何人ものプロ数学者の数十年の積み上げ成果だから、おれなんかがちょっと考えて解ける問題じゃないね
つづく 0398現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/11/19(日) 20:45:33.01ID:W1ZiI7BV>>397 つづき ＜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 This paper has been published in Gazette of the Australian Mathematical Society, Vol- ume 36, Number 5, November 2009, pp. 353{361. Received 29 February 2008; accepted for publication 6 October 2009. (抜粋)
ここに fν(x) ＝0 if x ∈ R - Q（無理数） ＝1/q^ν if x = p/q ∈ Q, irreducible （有理数で既約分数） で
Theorem 1. For ν > 2, the function fν is discontinuous (and consequently not differentiable) at the rationals, and continuous at the irrationals. With respect the differentiability, 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 un-countable. (引用終り)
