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.
proved the theorem for continuous functions, extended it to measurable functions, and extended it to arbitrary functions.
and 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 = –∞.
