On a theorem of Menkyna (Q1308864)

Let \(T_ d\) be the density topology of \(\mathbb R\) (\(\mathbb R\) -- the real line). The family \(T_{ae}\) of all sets \(A \in T_ d\) such that \(m(A - \text{int}\, A) = 0\) (\(m\) is the Lebesgue measure, \(\text{int}\,A\) denotes the Euclidean interior of \(A\)) is a topology said the a.e. topology [\textit{R. J. O'Malley}, Pac. J. Math. 72, 207--222 (1977; Zbl 0369.26003)]. A function \(f:(a,b) \to \mathbb R\), \((a,b) \subset \mathbb R\), is said to be a.e. continuous at a point \(x \in(a,b)\) if for every \(\varepsilon > 0\) there is a set \(B \in T_{ae}\) such that \(x \in B\) and \(f(B) \subset (f(x) - \varepsilon, f(x) + \varepsilon)\). The author gives a characterization of the set where an almost everywhere continuous Baire 1 function is a.e. continuous.