Denjoy-Young-Saks Theorem

The Denjoy-Young-Saks theorem deals with the characterization of the points where a real-valued function $f: [a, b] \rightarrow \mathbb{R}$ is differentiable in terms of its derivative. Specifically, the theorem characterizes the set of points where the function is non-differentiable in terms of the divergence of Dini's derivatives.

Here's a simplified version of the theorem:

Let $f: [a, b] \rightarrow \mathbb{R}$ be a real-valued function. Then, for almost every $x \in [a, b]$, one of the following holds:

1. $f$ is differentiable at $x$, and $f'(x)$ exists.
2. At least one pair of Dini's derivatives diverges at $x$, i.e., either the upper or lower Dini derivatives are infinite.

Dini's derivatives, defined for a function $f(x)$ at a point $x$, include four possible values:

1. $$D^+ f(x) = \lim_{{h \to 0^+}} \sup \frac{f(x+h) - f(x)}{h}$$
2. $$D^- f(x) = \lim_{{h \to 0^+}} \inf \frac{f(x+h) - f(x)}{h}$$
3. $$D_+ f(x) = \lim_{{h \to 0^-}} \sup \frac{f(x+h) - f(x)}{h}$$
4. $$D_- f(x) = \lim_{{h \to 0^-}} \inf \frac{f(x+h) - f(x)}{h}$$

"Almost every" typically means "except for a set of Lebesgue measure zero," in the context of real analysis.

The theorem is a generalization and sharpening of results by various mathematicians including Arnaud Denjoy, William Henry Young, and Saks, and provides a rigorous framework for understanding differentiability almost everywhere.