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DIFFERENTIABILITY OF A PATHOLOGICAL FUNCTION, DIOPHANTINE APPROXIMATION, AND A REFORMULATION OF THE THUE-SIEGEL-ROTH THEOREM JUAN LUIS VARONA 2009
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)
ここ、無理数を
(a) For every irrational number x with bounded elementsと、
(b) There exist infinitely many irrational numbers x such that fν is not differentiable at x.と
完全に２分したと読んだので、あとの測度論の下記Theorem 2
P6
Theorem 2. For ν > 2, let us denote
Cν = {f ∈ R : fν is continuous at x }
Dν = {f ∈ R : fν is differentiable 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)
つづく