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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394: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2017/12/22(金) 00:04:17.26 ID:UIwpFvOX

>>393 つづき

Note that as ε shrinks, the supremum of the function over the ball is monotone decreasing, so we have

lim sup _{x→ a}f(x)＝ ∈f _{ε >0}( sup {f(x):x ∈ E∩ B(a;ε )＼{a}})
and similarly
lim inf _{x→ a}f(x)＝ sup _{ε >0}( ∈f {f(x):x ∈ E∩ B(a;ε )＼{a}}).

This finally motivates the definitions for general topological spaces. Take X, Y, E and a as before, but now let X and Y both be topological spaces. In this case, we replace metric balls with neighborhoods:

lim sup _{x→ a}f(x)＝ ∈f { sup {f(x):x ∈ E∩ U＼{a}}:U open ,a ∈ U,E∩ U＼{a}≠ Φ }

lim inf _{x→ a}f(x)＝ sup { ∈f {f(x):x ∈ E∩ U＼{a}}:U open ,a ∈ U,E∩ U＼{a}≠ Φ }

(there is a way to write the formula using "lim" using nets and the neighborhood filter).
This version is often useful in discussions of semi-continuity which crop up in analysis quite often.
An interesting note is that this version subsumes the sequential version by considering sequences as functions from the natural numbers as a topological subspace of the extended real line, into the space (the closure of N in [?∞,∞], the extended real number line, is N ∪ {∞}.)
(引用終り)

以上

http://rio2016.5ch.net/test/read.cgi/math/1513201859/394

497: 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) 10:33:54.26 ID:Q5UHveEY

>>496 つづき

で、今回の「(a, b) 上でリプシッツ連続である」に関連する部分のみを、さらに抽出すると

[15] Gerald Arthur Heuer先生

THEOREM 4: The function f^2 is Lipschitzian but not
differentiable at the points of the set
{(1/2)*[m - sqrt(d)]: m is an integer
and there exists an integer n such that
d = m^2 - 4n is positive but not a perfect
square} . [This set is dense in the reals.]

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.

かな？

特に、THEOREM 5 変形トマエ函数（Ruler Function）のような、有理数で不連続、無理数で連続なる函数では、
”there is a dense uncountable subset of the reals at each point of which g fails to satisfy a Lipschitz condition.”
だと

だから、（A）”a dense uncountable subset”で、リプシッツ連続は満たさないは、実現できている

では、なぜ、（B）”内点を持たない閉集合の高々可算和”は、実現することができないのか？

[15] Gerald Arthur Heuer先生の（A）と、定理1.7 (422 に書いた定理)の（B）との差！

これを見極めない限り、素人の証明を読んでも仕方が無いと思う

まあ、年末は忙しい

ゆっくりやりましょう（＾＾

以上

http://rio2016.5ch.net/test/read.cgi/math/1513201859/497

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