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 outside a set of measure 0 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 = –∞.