On a Problem of Faure and Guénard

Abstract

In \cite{F17}, Faure and Gu\'enard put the following problem: Characterize the Denjoy$^*$-integrable functions $f:[a,b] \to \overline{{\mathbb R}}$ that can be approximated by two Baire $1$ functions $g_\epsilon$ and $h_\epsilon$, $\epsilon > 0$, that are ${\mathcal D}^*$-integrable. In the present article we show that this class of functions coincides with the class of all ${\mathcal D}^*$-integrable functions $f:[a,b] \to \overline{{\mathbb R}}$.