A concept of differential based on variational equivalence under generalized Riemann integration (Q1099270)

The concept of differential equivalence between set functions was introduced in 1930 by Kolmogorov. This notion has been reintroduced by Henstock in the sixties in the frame of his generalized Riemann integral. If S and T are two objects of integration, the corresponding equivalence relation is defined by the condition \(\int | S-T| =0.\) If \(S\simeq T\) and S is integrable, then the same is true for T and \(\int S=\int T\). The final object of integration is therefore not S itself but its equivalence class under the above relation. The aim of the paper is to show that these equivalence classes provide a viable mathematical formulation for the elusive concept of differential. The paper is written in the setting of the Kurzweil-Henstock integral of n-cells. It develops an introductory paper devoted to the special case of functions of one variable [the author, Am. Math. Mon. 93, 348-356 (1986; Zbl 0605.26007).