“Denjoy–Young–Saks theorem”的意思、由来-开放百科全书

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Denjoy–Young–Saks theorem

释义

Statement
References

In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere.

{{harvs|txt|last=Denjoy|authorlink=A. Denjoy|year=1915}} proved the theorem for continuous functions, {{harvs|txt|last=Young|authorlink=Grace Young|year=1917}} extended it to measurable functions, and {{harvs|txt|last=Saks|authorlink=S. Saks|year=1924}} extended it to arbitrary functions.{{harvtxt|Saks|1937|loc=Chapter IX, section 4}} and {{harvtxt|Bruckner|1978|loc=chapter IV, theorem 4.4}} give historical accounts of the theorem.

Statement

If f is a real valued function defined on an interval, then with the possible exception of a set of measure 0 on the interval, the Dini derivatives of f satisfy one of the following four conditions at each point:

f has a finite derivative
D⁺f = D_–f is finite, D⁻f = ∞, D₊f = –∞.
D⁻f = D₊f is finite, D⁺f = ∞, D_–f = –∞.
D⁻f = D⁺f = ∞, D_–f = D₊f = –∞.

References

{{Citation | last1=Bruckner | first1=Andrew M. | title=Differentiation of real functions | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-08910-0 | doi=10.1007/BFb0069821 | mr=507448 | year=1978 | volume=659}}
{{citation

| last = Saks
| first = Stanisław
| author-link =Stanisław Saks
| title = Theory of the Integral
| place = Warszawa-Lwów
| publisher = G.E. Stechert & Co.
| year = 1937
| series = Monografie Matematyczne
| volume = 7
| edition = 2nd
| pages = VI+347
| url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
| jfm = 63.0183.05
| zbl = 0017.30004
}}

{{Citation | last1=Young | first1=Grace Chisholm | title= On the Derivates of a Function | doi=10.1112/plms/s2-15.1.360 | year=1917 | journal= Proc. London Math. Soc. | volume=15 | issue=1 | pages=360–384}}

1 : Theorems in analysis

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