現代数学の系譜工学物理雑談古典ガロア理論も読む48

[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48 (625ﾚｽ)
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110: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/16(土) 14:36:58.33 ID:/2xvBEHK

>>98 関連

（>>35より、いままでと、重複もあるが、”co-meager”関連引用）
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.)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1513201859/110

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