現代数学の系譜工学物理雑談古典ガロア理論も読む48

[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48 (625ﾚｽ)
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506: 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) 20:59:06.39 ID:Q5UHveEY

>>497 関連

無理数で微分可能で、有理数のみ微分不可能という
函数の構成があったので、貼っておく（＾＾

http://www.mathcounterexamples.net/a-continuous-function-not-differentiable-at-the-rationals-differentiable-elsewhere/
ANALYSIS A CONTINUOUS FUNCTION NOT DIFFERENTIABLE AT THE RATIONALS BUT DIFFERENTIABLE ELSEWHERE NOVEMBER 30, 2014 JEAN-PIERRE MERX Math Counterexamples
(抜粋)
We build here a continuous function of one real variable whose derivative exists on R?Q and doesn’t have a left or right derivative on each point of Q.

As Q is (infinitely) countable, we can find a bijection n→rn from N to Q. We now reuse the function f defined here.
http://www.mathcounterexamples.net/a-differentiable-function-except-at-point-with-bounded-derivative
Recall f main properties:
略
This proves that hh is differentiable at aa with h′(a)=limn→+∞h′n(a). For a∈Q, we can find p∈N with a=rp.
Following a similar proof than above, the function lp:x→h(x)−up(x) is differentiable at a.
As f does not have left and right derivatives at 00, upup does not have left and right derivatives at a.
finally, the equality h=lp+up implies that hh also does not have left and right derivatives at a.

Conclusion:
the function h is differentiable at all irrational points but does not have left or right derivative at all rational points.
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1513201859/506

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