Denjoy–Young–Saks theorem
{{Short description|Mathematical theorem about Dini derivatives}}
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|author-link=A. Denjoy|year=1915}} proved the theorem for continuous functions, {{harvs|txt|last=Young|author-link=Grace Chisholm Young|year=1917}} extended it to measurable functions, and {{harvs|txt|last=Saks|author-link=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 = Warsaw-Lwów|publisher = G.E. Stechert & Co.|year = 1937|series = Monografie Matematyczne|volume = 7|edition = 2nd|url = http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl|archive-url=https://web.archive.org/web/20061212192909/http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl|archive-date=2006-12-12|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| url=https://zenodo.org/record/1447782/files/article.pdf }}
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