定理を引用しようとしたら、２CH（元５CH）では恰好悪いよ（＾＾ ↓ 定理を引用しようとしたら、２CH（現５CH）では恰好悪いよ（＾＾ 0472現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:24:03.24ID:OrUOLzdR>>470 "万に一つ、その定理と少なくとも証明が新しく、価値あるものなら、こんなところに書くのはもったいないよ（＾＾ ”（>>469） 0473現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:25:02.02ID:OrUOLzdR その定理が正しい確率を直観で表わしたんだが？（＾＾ 0474現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE 2017/12/09(土) 23:32:59.79ID:OrUOLzdR>>443に有る（英文）定理では ”Let g be continuous and discontinuous on sets of points that are each dense in the reals. ”とある
１．Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.（>>285より） w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数) 無理数で0。ついでに、f_w(0) = 1 （>>285より。＊） （「無理数で、リプシッツ連続」は>>284以下の既出文献でさんざん証明＊＊）済みで略す） ２．f_w(p/q) = 1/w(q)＞0と出来るとして、p/q（有理数）では、不連続になる。（自明だが念のために書いた） ３．このRuler Function に、新定理が適用可能とする。 ４．R−B_f ⊂ Q ＝ ∪[p∈Q] { p } …(1) (1)の右辺は疎な閉集合の可算和だから、上の新定理が使えて、f はある開区間(a,b)の上でリプシッツ連続になる。
？ この後、そのままで良いのか？
特に、(a,b)の上で連続になる。QはR上で稠密だから、x∈(a,b)∩Qが取れる。 fは点xで不連続であるが、しかし(a,b)の上で連続に、矛盾する。 QED
ａ）なので、”このRuler Function に、新定理が適用可能”がおかしいか ｂ）新定理がおかしいか
注 ＊）f_w(0) = 1を書く意味は、0は無理数でもなく、p/qとも表せないということかな ＊＊）（>>285より抜粋） ** For r = 2, f^r is nowhere differentiable and satisfies a pointwise Lipschitz condition on a set that is dense in the reals. Heuer [15]
** For r > 2, f^r is differentiable on a set whose intersection with every open interval has Hausdorff dimension 1 - 2/r. Frantz [20]
** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25) (引用終り)
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
これを読んでいて、疑問に思ったことが２点ある
１． ”[20] Marc Frantz, "Two functions whose powers make fractals", American Mathematical Monthly 105 #7 (Aug./Sept. 1998), 609-617. [MR 99g：28018； Zbl 952.28006] Following up on Darst/Taylor [18] above, Frantz investigates the Hausdorff dimension of the graphs of f^r.
THEOREM 1： If r > 2, then the Hausdorff dimension of the non-differentiability set for f^r is 2/r.”
一方、 ”[18] Richard Brian Darst and Gerald D. Taylor, "Differentiating Powers of an Old Friend", American Mathematical Monthly 103 #5 (May 1996), 415-416. [MR1400724； Zbl 861.26002]
Define f：R --> R by f(x) = 0 if x is irrational or zero, and f(p/q) = 1/q for p,q relatively prime with q > 0. They note that the set of points at which f is not continuous is the set of nonzero rational numbers.
THEOREM： If 1 < r <= 2, then f^r is differentiable only at zero. If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).”
だから、[18] からすると、If r > 2, then f^r is differentiable almost everywhere (Lebesgue measure).→Hausdorff dimension =1 で、"1 - 2/r（>>285）"ではないのでは？
２． ”Let g be continuous and discontinuous on sets of points that are each dense in the reals.” とは、continuous, discontinuous, 両者とも、Hausdorff dimension =1/2 見たいな形で、お互いが混じり合っているイメージなんだけど、おかしいかな？ で、無理数と有理数だと、前者がHausdorff dimension =1、後者がHausdorff dimension =0 なんだけど・・・ 「函数の連続点の全体からなる集合は開集合の可算個の交わり（Gδ-集合）である。また不連続点の全体は閉集合の可算個の合併（Fσ-集合）である。」（by 上記wikipedia 不連続性の分類 ） だから、それで良いのか・・な（＾＾
Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.（>>285より） w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)
は、おまえの新定理の反例になってないか？
１．（>>481 wikipediaより）「不連続点の全体は閉集合の可算個の合併（Fσ-集合）である」を認めるとする ２．”** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)”（>>285より） Hausdorff dimension zero → 個々の不連続点の閉集合は、R上長さを持たない、つまり、”内点を持たない”が言えると思う（未証明だが） ３．とすると、その定理の”R−B_f が高々可算無限個の疎な閉集合の和で被覆できる”が言えるだろ？ ４．で、R−B_f は疎な閉集合の可算和だから、新定理が使えて、f はある開区間(a,b)の上でリプシッツ連続になる。 ５．で、特に、(a,b)の上で連続になる。QはR上で稠密だから、x∈(a,b)∩Qが取れる。fは点xで不連続であるが、しかし(a,b)の上で連続に、矛盾する。
まあ、要するに、この”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.”（>>285より）というのは ” be continuous and discontinuous on sets of points that are each dense in the reals.”（>>285より）が、実現された関数なわけだ
＞１．（>>481 wikipediaより）「不連続点の全体は閉集合の可算個の合併（Fσ-集合）である」を認めるとする ＞２．”** f_w is differentiable on a set whose complement has Hausdorff dimension zero. Jurek [4] (pp. 24-25)”（>>285より） ＞ Hausdorff dimension zero → 個々の不連続点の閉集合は、R上長さを持たない、つまり、”内点を持たない”が言えると思う（未証明だが） ＞３．とすると、その定理の”R−B_f が高々可算無限個の疎な閉集合の和で被覆できる”が言えるだろ？
１点、（>>497）”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.（>>285より） w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)” が、反例になるだろうと指摘した
１． （>>534 より）”１点、（>>497）”Ruler Function f_w(p/q) = 1/w(q) where p and q are relatively prime integers.（>>285より） w(q) an increasing function that eventually majorizes every power function. (いかなるq^rよりも急増加関数)” が、反例になるだろうと指摘した それ、>>284-285に出典が上がっているだろ？ あなたのすべきことは、私への反論でなく、出典に当たって、本当に反例かそうでないか、直接確かめることじゃないのか？” と書いたけど、相変わらず原典に当たってないでしょ？ あなたの反例でないという理屈が、単におれが挙げた理由付けについての反論に過ぎないでしょ？
それから、反例の原典に当たったと言うけれど、おれの言っている原典は、単に下記URLを覗くだけじゃなく、その元の引用文献に当たれってことだ！ 例えば、下記[4][13]や、 H. M. Sengupta and B. K. Lahiri, "A note on derivatives of a function",Bulletin of the Calcutta Mathematical Society 49 (1957)
それに、[15] や、Edward Maurice Beesley, Anthony Perry Morse, and Donald Chesley Pfaff, "Lipschitzian points", American Mathematical Monthly 79 #6 (June/July 1972) くらいは、これは見とかないとね
（例えば、”(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), ”とかに関連した部分など。 あんたみたく、簡単に証明できるというなら、Heuer先生"We omit the proof, because it is rather lengthy”とは書かないだろう・・）
[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.
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.
[15] Gerald Arthur Heuer, "Functions continuous at the irrationals and discontinuous at the rationals", American Mathematical Monthly 72 #4 (April 1965), 370-373. [MR 31 #3550; Zbl 131.29201]
THEOREM 5: If g is a function discontinuous at the rationals and continuous at the irrationals, then there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.
(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. (引用終り)
