Denjoy–Young–Saks theorem

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.

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), Dold, A.; Eckmann, B. (eds.), Differentiation of Real Functions, Lecture Notes in Mathematics, vol. 659, Berlin, Heidelberg, 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. VII, English translation by L. C. Young and two additional notes by S. Bannach (Second Revised ed.), New York: Hafnar Publishing Company, 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