On approximate derivatives and Krzyzewski-Foran lemma. (English) Zbl 1059.26004

Let \(E\subset \mathbb{R}\) and \(g:E\to \mathbb{R}\). The author shows that if \(| g(E)| = 0\), then the inequalities \(g_{\text{ap}}(x)\leq 0\leq \overline{g}_{\text{ap}}(x)\) hold almost everywhere on \(E\). This result generalizes results of K. Krzyzewski and J. Foran. The author also improves the chain rule for approximate derivatives [J. Foran, “A chain rule for the approximate derivative and change of variables for the \(D\)-integrals”, Real Anal. Exch. 8, 443–454 (1983; Zbl 0548.26003)] and obtains necessary and sufficient conditions for its validity almost everywhere.


26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A48 Monotonic functions, generalizations
26A46 Absolutely continuous real functions in one variable
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence


Zbl 0548.26003
Full Text: DOI