ガロア第一論文と乗数イデアル他関連資料スレ17

[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ17 (1002ﾚｽ)
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432: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/05/17(土) 20:00:54.76 ID:y2zepp9J

>>427 補足
>（”一様連続”の条件を外せるかは ちょっと保留）

”一様連続”を仮定するのが、良さそうだね
下記の通り
一様連続
→Uniform continuity（英文情報（圧倒的に良質情報が多い））
→Cauchy continuity（For a function between metric spaces, uniform continuity implies Cauchy continuity (Fitzpatrick 2006). ）
→Cauchy-continuous function Examples and non-examples
と辿れる
ここで Q上 Cauchy-continuou関数だが Uniform continuouでない関数が、 non-example として構成されている（下記）
こいつは Rへ連続関数として延長不可だ！ｗ ；ｐ）

(参考)
https://ja.wikipedia.org/wiki/%E4%B8%80%E6%A7%98%E9%80%A3%E7%B6%9A
一様連続

https://en.wikipedia.org/wiki/Uniform_continuity
Uniform continuity

Other characterizations
Cauchy continuity
For a function between metric spaces, uniform continuity implies Cauchy continuity (Fitzpatrick 2006).
（注：逆は不成立（下記））

Relations with the extension problem
A sufficient condition for
f to extend to a continuous function
f:X→R is that it is Cauchy-continuous, i.e., the image under
f of a Cauchy sequence remains Cauchy. If
X is complete (and thus the completion of S), then every continuous function from
X to a metric space Y is Cauchy-continuous. Therefore when
X is complete, f extends to a continuous function
f:X→R if and only if f is Cauchy-continuous.
It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to X.

https://en.wikipedia.org/wiki/Cauchy-continuous_function
Cauchy-continuous function

Examples and non-examples
Since the real line
R is complete, continuous functions on R are Cauchy-continuous. On the subspace
Q of rational numbers, however, matters are different. For example, define a two-valued function so that
f(x) is
0 when x^2 is less than
2 but 1 when x^2 is greater than 2.
(Note that x^2 is never equal to 2 for any rational number x.)
This function is continuous on Q but not Cauchy-continuous, since it cannot be extended continuously to R.
{\displaystyle \mathbb {R} .} On the other hand, any uniformly continuous function on Q must be Cauchy-continuous.
For a non-uniform example on Q, let f(x) be 2^x;
this is not uniformly continuous (on all of Q),
but it is Cauchy-continuous. (This example works equally well on R.)
A Cauchy sequence (y1,y2,…) in Y can be identified with a Cauchy-continuous function from
{1,1/2,1/3,…} to Y, defined by f(1/n)=yn.
If Y is complete, then this can be extended to {1,1/2,1/3,…}; f(x) will be the limit of the Cauchy sequence.

http://rio2016.5ch.net/test/read.cgi/math/1746597368/432

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