Denjoy-Saks-Young Theorem

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

1. There is a finite derivative,

2. and are finite and equal, , and ,

3. and are finite and equal, , and , or

4. and .

Here, , , , and denote the upper right, lower right, upper left, and lower left Dini derivatives of , respectively.