現代数学の系譜工学物理雑談古典ガロア理論も読む48

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35: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/14(木) 22:44:02.12 ID:oVKNFyGV

>>29
どうも。スレ主です。
ご指摘レスありがとう

ところで、どういう意味かな？
「ぷふ」さんの「確かに有理数で不連続無理数で微分可能な関数は存在しないですね」というのは
>>21に書いてある命題Aのことでしょ

でそれは、前スレ284-285 に有るとおり、上記>>20の証明の前（2006以前）に、プロ数学者が命題Aは得ているよ
（再度引用しておく）
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)
Using ruler-like functions that "damp-out" quicker
than any power of f gives behavior that one would
expect from the above.

Let w:Z+ --> Z+ be an increasing function that
eventually majorizes every power function. Define
f_w(x) = 0 for x irrational, f_w(0) = 1, and
f_w(p/q) = 1/w(q) where p and q are relatively
prime integers.

** f_w is differentiable on a set whose complement
has Hausdorff dimension zero. Jurek [4] (pp. 24-25)

Interesting, each of the sets of points where these
functions fail to be differentiable is large in the
sense of Baire category.

THEOREM: Let g be continuous and discontinuous on sets
of points that are each dense in the reals.
Then g fails to have a derivative on a
co-meager (residual) set of points. In fact,
g fails to satisfy a pointwise Lipschitz
condition, a pointwise Holder condition,
or even any specified pointwise modulus of
continuity condition on a co-meager set.

(Each co-meager set has c points in every interval.)
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1513201859/35

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