On the difference property of the family of functions with the Baire property (Q1878578)

A class of functions \({\mathcal F}\) is said to have the difference property if every function for which the difference function \( x \mapsto f(x+h) - f(x) \) \((x \in {\mathbb R})\) belongs to \({\mathcal F}\) for every \( h \in {\mathbb R} \) is of the form \( f = g + A \) where \(g\) belongs to \({\mathcal F}\) and \(A\) is additive. We say that a function \(f:\mathbb R\to\mathbb R\) has the Baire property if, for every open set \( U \subset \mathbb R\), \( f^{-1} (U) \) is the symmetric difference of an open set and a meager set. It is known that the family of functions with the Baire property does not have the difference property under Continuum Hypothesis. According to the main result of this paper, it is consistent with ZFC that the family of functions with the Baire property has the difference property. The research is motivated by an analogous result of \textit{M. Laczkovich} [Real Anal. Exch. 24, No.~2, 663--676 (1998; Zbl 0964.03056)] concerning the class of Lebesgue measurable functions. The author notices that the main theorem has been independently obtained by \textit{T. Mátrai} [``Weak difference property of functions with the Baire property'', manuscript].