mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Denjoy–Young–Saks theorem gives some possibilities for the

Dini derivative In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini ...

s of a function that hold

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

. proved the theorem for

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...

s, extended it to

measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...

s, 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

* * * {{DEFAULTSORT:Denjoy-Young-Saks theorem Theorems in analysis