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, Chapter IX, section 4) and Bruckner (1978, chapter IV, theorem 4.4) give historical accounts of the theorem.
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 = –∞.
- Bruckner, Andrew M. (1978), Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Berlin, New York: Springer-Verlag, doi: 10.1007/BFb0069821, ISBN 978-3-540-08910-0, MR 0507448
- Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warszawa- Lwów: G.E. Stechert & Co., JFM 63.0183.05, Zbl 0017.30004, archived from the original on 2006-12-12
- Young, Grace Chisholm (1917), "On the Derivates of a Function" (PDF), Proc. London Math. Soc., 15 (1): 360–384, doi: 10.1112/plms/s2-15.1.360
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, Chapter IX, section 4) and Bruckner (1978, chapter IV, theorem 4.4) give historical accounts of the theorem.
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 = –∞.
- Bruckner, Andrew M. (1978), Differentiation of real functions, Lecture Notes in Mathematics, vol. 659, Berlin, New York: Springer-Verlag, doi: 10.1007/BFb0069821, ISBN 978-3-540-08910-0, MR 0507448
- Saks, Stanisław (1937), Theory of the Integral, Monografie Matematyczne, vol. 7 (2nd ed.), Warszawa- Lwów: G.E. Stechert & Co., JFM 63.0183.05, Zbl 0017.30004, archived from the original on 2006-12-12
- Young, Grace Chisholm (1917), "On the Derivates of a Function" (PDF), Proc. London Math. Soc., 15 (1): 360–384, doi: 10.1112/plms/s2-15.1.360