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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496: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/24(日) 10:31:48.31 ID:Q5UHveEY

>>494 つづき

(p. 373) "We omit the proof, because it is rather lengthy,
and one would hope to generalize the theorem by replacing
the rationals by an arbitrary dense set, and possibly to
show that the set of points at which g fails to be
Lipschitzian is a residual set."

NOTE: Sengupta/Lahiri had essentially obtained this result
in 1957 (the points of discontinuity have to form an
F_sigma set, however). See my remark in [13] above.
This result is also proved in Gerald Arthur Heuer,
"A property of functions discontinuous on a dense set",
American Mathematical Monthly 73 #4 (April 1966),
378-379 [MR 34 #2791]. Heuer proves that for each
0 < s <= 1 and for each f:R --> R such that
{x: f is continuous at x} is dense in R and
{x: f is not continuous at x} is dense in R,
the set of points where f does not satisfy a
pointwise Holder condition of order s is the
complement of a first category set (i.e. a co-meager
set). By choosing s < 1, we obtain a stronger version
of Sengupta/Lahiri's result. By intersecting the
co-meager sets for s = 1/2, 1/3, 1/4, ..., we get
a co-meager set G such that, for each x in G, f does
not satisfy a pointwise Holder condition at x for
any positive Holder exponent. (Heuer does not
explicitly state this last result.) A metric space
version of Heuer's result for an arbitrary given
pointwise modulus of continuity condition is essentially
given in: Edward Maurice Beesley, Anthony Perry Morse,
and Donald Chesley Pfaff, "Lipschitzian points",
American Mathematical Monthly 79 #6 (June/July 1972),
603-608 [MR 46 #304; Zbl 239.26004]. See also the last
theorem in Norton [17] below.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1513201859/496

526: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/25(月) 07:58:39.31 ID:R/y0B5bE

>>521-522
>>カントール集合で``1個''です
>”S ⊆ ∪iFi”で、Sは集合濃度で連続まで許すのか？
>当然ですよ

なんだよ（＾＾
早く言ってくれればよかったのに（＾＾

でな、下記

リウヴィル数は、非可算集合、実数内で稠密で、ルベーグ測度は 0 であるから、内点を持たない
リウヴィル数の各点は、閉集合だと思うが、それで良いかな？

で、いま問題のRuler Functionでは、リウヴィル数が鬼門で
”not Lipschitzian at the Liouville numbers, for every r > 0”なんだよ

つまり、r→∞にしても、リウヴィル数以外の無理数は、Lipschitzianになるが、at the Liouville numbersではだめだと
で、そうすると、定理1.7 (422 に書いた定理)の反例になってないか？

(>>151)
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%82%A6%E3%83%B4%E3%82%A3%E3%83%AB%E6%95%B0
リウヴィル数
(抜粋)
・リウヴィル数全体からなる集合は非可算集合であり、実数内で稠密であるが、1次元ルベーグ測度は 0 である。

http://mathforum.org/kb/message.jspa?messageID=5432910 （>>35より）
Topic: Differentiability of the Ruler Function Dave L. Renfro Posted: Dec 13, 2006 Replies: 3 Last Post: Jan 10, 2007
(>>494)
(抜粋)
THEOREM 2: The function f^r is: (B) continuous but not Lipschitzian at the Liouville numbers, for every r > 0;

(>>492）
(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.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1513201859/526

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