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Denjoy–Young–Saks theorem
Mathematical theorem about Dini derivatives
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. 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 = –∞.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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