[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)
上下前次1-新
抽出解除 ﾚｽ栞
このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索 歴削→次ｽﾚ 栞削→次ｽﾚ 過去ﾛｸﾞﾒﾆｭｰ
ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡
238(1): １３２人目の素数さん [sage] 2018/01/07(日) 18:24:19.00 ID:WplCTiTS(2/4) AAS
>>236
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。
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)
上下前次1-新書関写板覧索設栞歴
ｽﾚ情報 赤ﾚｽ抽出 画像ﾚｽ抽出 歴の未読ｽﾚ AAｻﾑﾈｲﾙ
ぬこの手 ぬこTOP 0.042s