現代数学の系譜工学物理雑談古典ガロア理論も読む49

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281: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2018/01/08(月) 21:10:58.54 ID:KgoytC9i

>>280 つづき

ところで、下記は、指示関数そのものではないが、R中の部分集合Bfとその補集合R−Bfに分けて、関数値を決めていると考えることができる
(>>268)
https://kbeanland.files.wordpress.com/2010/01/beanlandrobstevensonmonthly.pdf
Modifications of Thomae’s function and differentiability, (with James Roberts and Craig Stevenson) Amer. Math. Monthly, 116 (2009), no. 6, 531-535.
の記載より(抜粋)
2. MODIFIED THOMAE FUNCTION.
Let (ai) be a sequence of reals decreasing to zero. Define the modified Thomae
function with respect to (ai) as follows:
T(ai)(x)
= 0 if x ∈ R ＼ Q,
= an if x = m/n where m and n are coprime,
= 1 if x = 0.

Since limn an = 0, T(an) is continuous on the irrationals. The faster the sequence (ai)
tends to zero, the larger the set of irrationals on which T(ai) will be differentiable.

3. A DENSE SET. While attempting to prove that T(1/n^2) is differentiable on the irrationals,
we discovered that quite the opposite is actually true. In fact, as the following
proposition indicates, functions that are zero on the irrationals and positive on the rationals
will always be non-differentiable on a rather large set.

Proposition 3.1. Let f be a function on R that is positive on the rationals and 0 on the irrationals.
Then there is an uncountable dense set of irrationals on which f is not differentiable.
(引用終り)

つづく

http://rio2016.5ch.net/test/read.cgi/math/1514376850/281

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