Symmetric derivates, scattered, and semi-scattered sets

Abstract:

We call a set right scattered (left scattered) if every nonempty subset contains a point isolated on the right (left). We establish the following monotonicity theorem for the symmetric derivative. If a real function $f$ has a nonnegative lower symmetric derivate on an open interval $I$, then there is a nondecreasing function $g$ such that $f(x) > g(x)$ on a right scattered set and $f(x) < g(x)$ on a left scattered set. Furthermore, if $R$ is any right scattered set and $L$ is any left scattered set disjoint with $R$, then there is a function which is positive on $R$, negative on $L$, zero otherwise, and which has a zero lower symmetric derivate everywhere. We obtain some consequence including an analogue of the Mean Value Theorem and a new proof of an old theorem of Charzynski.