# Some resources for the upper (lower) derivative

Lately I have been looking a bit into Perron integral. This integral is defined using lower and upper derivative: $$\underline Df(x) =\liminf_{y\to x} \frac{f(y)-f(x)}{y-x} \qquad \overline Df(x)=\limsup_{y\to x} \frac{f(y)-f(x)}{y-x}.$$ (To be more specific, when defining Perron integral of a given function $g$, we are looking at functions $u$, $v$ such that $\overline Dv \le g \le \underline Du$ and the functions $u$, $v$ are, in some sense, close to each other.)

So far it seems that the text I am reading only develops the properties of the upper/lower derivative that are needed in the proofs of various properties of Perron integral. It would be useful to me to gain a bit more insight into these notions, so that I can understand their usage in connection with this integral a bit better. Moreover, these derivatives seem to be interesting by themselves. Of course, I could simply experiment a bit, try to find upper/lower derivatives of some functions and maybe even discover some of their properties of $\underline Df$ and $\overline Df$ by myself. Still, if there are some goods texts which study them more in detail, it would be nice to know about them.

TL;DR: Can you recommend some texts (books, lecture notes) which develop some basic properties of lower and upper derivative? (It would be even better if they have some examples and exercises.)

• See Gordon Russell A., The Integrals of Lebesgue, Denjoy, Perron, and Henstock (Graduate Studies in Mathematics, Vol.4), American Mathematical Society (1994), chapter 4. – Nosrati Nov 28 '17 at 18:15