Dini derivative

Summary

The Dini derivatives of a real function at a point are the four one-sided upper and lower limits of its difference quotients. Unlike the derivative, they always exist, possibly as $+\infty$ or $-\infty$. A function is differentiable at a point exactly when its four Dini derivatives there are equal and finite. They are the main tool for proving differentiability almost everywhere, as in Lebesgue's theorem on monotone functions.