A change of variables formula for Darboux integrals (Q1978977)

The formula \[ \overline {\int _{g(a)}^{g(b)}}f(x) dx \leq \overline {\int _a^b}f(g(t)) Dg(t) dt \] is proved for Darboux integrals. Here \(Dg\) is one of the Dini derivates, \(f\) and \(Dg\) are assumed to be bounded, and at a.e.\ point of \([a,b]\) at least one of the functions \(f(g(\cdot))\) or \(Dg(\cdot)\) is assumed to be continuous. As a consequence, the change of variables formula for the Riemann integration \[ \int_{g(a)}^{g(b)}f(x) dx =\int _a^bf(g(t)) Dg(t) dt \] is established under various hypotheses.