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Denjoy-Saks-Young Theorem

Let f be a finite real-valued function defined on an interval [a,b]. Then at every point in [a,b] except on a set of Lebesgue measure zero, either:

1. There is a finite derivative,

2. D^+f and D_f are finite and equal, D^-f=+infty, and D_+f=-infty,

3. D^-f and D_+f are finite and equal, D^+f=+infty, and D_-f=-infty, or

4. D^+f=D^-f=+infty and D_+f=D_-f=-infty.

Here, D^+f, D_+f, D^-f, and D_-f denote the upper right, lower right, upper left, and lower left Dini derivatives of f, respectively.

See also

Derivative, Dini Derivative, Interval, Lebesgue Measure, Lower Left Dini Derivative, Lower Right Dini Derivative, Measure Zero, Upper Left Dini Derivative, Upper Right Dini Derivative

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References

Kannan, R. and Krueger, C. K. Advanced Analysis on the Real Line. New York: Springer-Verlag, 1996.

Cite this as:

Weisstein, Eric W. "Denjoy-Saks-Young Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Denjoy-Saks-YoungTheorem.html

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