Denjoy–Young–Saks theorem
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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. Denjoy (1915) proved the theorem for continuous functions, Young (1917) extended it to measurable functions, and Saks (1924) extended it to arbitrary functions. (Saks 1937) and (Bruckner 1978) 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
- Bruckner, Andrew M. (1978), Differentiation of real functions, Lecture Notes in Mathematics, 659, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069821, ISBN 978-3-540-08910-0
- Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, 7 (2nd ed.), Warszawa-Lwów: G.E. Stechert & Co., http://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez=pl
- Young, Grace Chisholm (1917), "On the Derivates of a Function", Proc. London Math. Soc. 15 (1): 360–384, doi:10.1112/plms/s2-15.1.360, https://zenodo.org/record/1447782/files/article.pdf
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