[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49 (658ﾚｽ)
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41(4): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/28(木) 23:44:24.28 ID:IsA0R4yK(6/8) AAS
>>40 つづく
[13] Gerald Arthur Heuer, "Functions continuous at irrationals and discontinuous at rationals", abstract of talk given 2 November 1963 at the annual fall meeting of the Minnesota Section of the MAA, American Mathematical Monthly 71 #3 (March 1964), 349.
The complete text of the abstract follows, with minor editing changes to accommodate ASCII format.
Earlier results of Porter, Fort, and others suggest additional questions about the functions in the title. Differentiability and Lipschitz conditions are considered. Special attention ispaid to the ruler function (f) and its powers.
Sample results:
THEOREM:
If 0 < r < 2, f^r is nowhere Lipschitzian; f^2 is nowhere differentiable, but is Lipschitzian on a dense subset of the reals.
THEOREM:
If r > 0, f^r is continuous but not Lipschitzian at every Liouville number;
if r > 2, f^r is differentiable at every algebraic irrational.
THEOREM:
If g is continuous at the irrationals and not continuous at the rationals, then there exists a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
REMARK BY RENFRO:
The last theorem follows from the following stronger and more general result.
Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R.
Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.
Then E is co-meager in R (i.e. the complement of a first category set).
This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function",
Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below.
(引用終り)
つづく
42(3): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2017/12/28(木) 23:46:02.50 ID:IsA0R4yK(7/8) AAS
>>41 つづき
ああ、いま改めて読むと
Bulletin of the Calcutta Mathematical Society 49 (1957) Senguptaより
”・・・ f is continuous and discontinuous are each dense in R.
Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.
Then E is co-meager in R (i.e. the complement of a first category set).”
なんてありますね。”at least one of the four Dini derivates of f is infinite”が、貴方の定理に近いかな？
”Then E is co-meager in R (i.e. the complement of a first category set).”か・・
これか、これに近い文献を読まないことには、訳わからんな
えーと、Meagre setか・・
”E is co-meager in R”が、イメージできんな・・（＾＾
前提a)(連続不連続が稠密)を、b)(連続とディニ微分発散が稠密な組み合わせ)に、緩和しても・・
a) f is continuous and discontinuous are each dense in R.
↓
b) f is continuous and the E *) are each dense in R. ( *)the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.)
a)Eは、co-meager
↓
b)Eは、meager
には出来ない？ それとも出来るの？
定理1.7成立なら、「 meager には出来ない」？
これ、やっぱり元論文読まないと、イメージ湧かないな〜（＾＾
まあ、ゆっくりやろうや
45(3): １３２人目の素数さん [sage] 2017/12/29(金) 00:12:33.08 ID:gcYWyS10(1/7) AAS
>>40-42
＞まだ、疑問に思っているのは
＞下記のDifferentiability of the Ruler Functionの記述と貴方の定理との整合性だ
悪あがきは やめたまえ。前スレ540 で既に述べたとおり、
スレ主の大好きな f^r と f_w は、例の定理の反例になり得ない。
2chｽﾚ:math
このレスにより、R−B_{f^r} は第一類集合にならず、R−B_{f_w} も第一類集合にならないので、
f^r と f_w は例の定理の「適用範囲外」ということになり、よって例の定理の反例になり得ない。
また、f^r と f_w が例の定理の「適用範囲外」であるという事実により、Differentiability of the Ruler Function が
どのように記述されていようとも、そのことと例の定理との間の整合性なんか 全 く 考 え る 必 要 が な い 。
189(4): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/05(金) 20:13:29.17 ID:miqaDy4s(11/12) AAS
>>188
おっちゃん、どうも、スレ主です。
レスありがとう（＾＾
（>>180より）
”定理1.7 (422 に書いた定理)
f : R → R とする．
Bf :={x ∈ R ｜ lim sup y→x ｜（f(y) − f(x)）/（y − x）｜< +∞ }
と置く: もしR−Bf が内点を持たない閉集合の高々可算和で被覆できるならば、 f はある開区間(a, b) の
上でリプシッツ連続である．”
この定理1.7の面白さは
”系1.8 有理数の点で不連続, 無理数の点で微分可能となるf : R → R は存在しない.”（>>184）
を著しく拡張しているところだ
つまり、系1.8において、
１）不連続→リプシッツ連続でない
２）微分可能→リプシッツ連続
３）稠密：有理数と無理の稠密性→もっと一般な稠密性（但し、片方は可算無限濃度限定）
の３つの特性で、系1.8を拡張したものが定理1.7になっているってこと
これに匹敵する結果は、>>41-42に書いたが
”Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R.
Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.
Then E is co-meager in R (i.e. the complement of a first category set).
This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function",
Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. ”
つまり、一般な稠密性（但し、H. M. Sengupta and B. K. Lahiriは、可算非可算に関係なく）
”the sets of points at which f is continuous and discontinuous are each dense in R.”なのだが
しかし、この discontinuous →リプシッツ連続でないという、上記１）の特性で、定理1.7は拡張されているのだ
そこが、この定理1.7の面白さであり、斬新さだ
成り立てばだがね（＾＾
245(2): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [sage] 2018/01/07(日) 20:46:08.89 ID:2l42E8SE(23/29) AAS
>>238
おっちゃん、どうも、スレ主です。
>いっておくけど、系 1.8 の結果は
>有理数の点で不連続, 無理数の点で一様連続となるf : R → R は存在しない
>まで拡張出来る。
ああ、下記だな”g fails to satisfy・・even any specified pointwise modulus of continuity condition on a co-meager set.”＆
”Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.
Then E is co-meager in R (i.e. the complement of a first category set).”
繰返すが、
”any specified pointwise modulus of continuity condition” ＆ ”at least one of the four Dini derivates of f is infinite”
だから（特に後者Dini微分）、どこかの無理数の点で一様連続も破綻するだろうな
（>>40より）http://mathforum.org/kb/message.jspa?messageID=5432910
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.)
（>>41より）
REMARK BY RENFRO:
The last theorem follows from the following stronger and more general result.
Let f:R --> R be such that the sets of points at which f is continuous and discontinuous are each dense in R.
Let E be the set of points at which f is continuous and where at least one of the four Dini derivates of f is infinite.
Then E is co-meager in R (i.e. the complement of a first category set).
This was proved in H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function",
Bulletin of the Calcutta Mathematical Society 49 (1957), 189-191 [MR 20 #5257; Zbl 85.04502]. See also my note in item [15] below.
(引用終り)
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