Level sets of derivatives

Abstract:

The main result of the paper is the characterization of those triples $S$, $G$ and $E$ of subsets of the reals for which there exists an everywhere differentiable real-valued function $f$ of one real variable such that $E = \{ x;f’(x) > 0\}$, $G = \{ x;f’(x) = + \infty \}$ and $S$ is the set of those points of $E$ at which $f$ is discontinuous. This description is formulated with the help of a certain density-type property of subsets of the reals (called property $({\text {Z))}}$ introduced in the paper. The main result leads to a complete description of the structure of the sets $\{ x;f’(x) > 0\}$ and $\{ x;f’(x) = 0\}$ for three most important classes of functions $f:$ finitely differentiable functions, continuous differentiable functions and everywhere differentiable functions. (A complete description of the structure of these sets for the class of Lipschitz, everywhere differentiable functions was given by Zahorski in his fundamental paper [22].) The connection of these results with Zahorski’s classes ${M_2}$, ${M_3}$ and ${M_4}$ is discussed.