>>17

これ（下記）ちょっと読み直しているのだが、
下記の”Peano differentiable”、 ”Peano derivative”がよく分らない
検索したが、
分り易い文献がヒットせず（＾＾；
＜The modified ruler function のまとめサイト下記＞
http://mathforum.org/kb/message.jspa?messageID=5432910
Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007
(抜粋)
[17] Alec Norton [Kercheval], "Continued fractions and
differentiability of functions", American Mathematical
Monthly 95 #7 (Aug./Sept. 1988), 639-643.
[MR 89j:26009; Zbl 654.26006]

Define g:R --> R by g(x) = 0 if x is irrational, g(0) = 1,
and g(p/q) = 1/2^q if p and q are relatively prime with
q > 0.

PROPOSITION: There exists a partition A_0, A_1, A_2, ...
and A_oo of the irrational numbers, where each set is
c-dense in the reals, such that g is infinitely Peano
differentiable at each point of A_oo and, for each
n >= 0 and for each x in A_n, g is n-times Peano
differentiable but not (n+1)-times Peano differentiable
at x. Moreover, the complement of A_0 is a first
category set and the complement of A_oo is a Lebesgue
measure zero set.

NOTE: Norton says "uncountable dense sets" instead of
"c-dense in the reals". While it is a little
ambiguous what he means (uncountable sets that
are dense in the reals, or sets having an uncountable
intersection with every open interval) until one
gets to the proof, it is clear from the proof
(the sets involved are Borel, for instance)
that the sets are, in fact, c-dense in the reals.

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