A note on SAC property (Q2774496)

The paper is related to the following open problem. A function \(f: \mathbb R \to \mathbb R\) is said to have SAC property if for every \(\eta >0\) there is an approximately continuous function \(r:\mathbb R \to \mathbb R_+\) such that for each \(x\) and \(h\) the inequality \(0<|h|<r(x)\) implies NEWLINE\[NEWLINE\biggl |\frac 1h \int _x^{x+h} f(t) dt - f(x)\biggr |<\eta NEWLINE\]NEWLINE [see \textit{Z. Grande} ``On equi-derivatives'', Real Anal. Exch. 21, No. 2, 637-647 (1995; Zbl 0879.26020)]. The problem is: does every bounded approximately continuous function have SAC property? NEWLINENEWLINENEWLINEThe authors claim that the following two results presented in the paper might lead to a positive answer to the problem: NEWLINENEWLINENEWLINE(1) Given an approximately continuous function \(f\) the function NEWLINE\[NEWLINEp(x)=\inf \Biggl \{ |h|:\biggl |\frac 1h \int _x^x f(t) dt - f(x) \biggr |> \eta \Biggr \}NEWLINE\]NEWLINE is measurable. NEWLINENEWLINENEWLINE(2) If \(p\) is approximately lower semicontinuous of Baire class one and \(p>0\) then there is an approximately continuous function \(r\) with \(0<r\leq p\). NEWLINENEWLINENEWLINEReviewer remark. The latter result does not seem to be new [see \textit{J. Lukes, J. Malý} and \textit{L. Zajícek}, ``Fine topology methods in real analysis and potential theory'' (1986; Zbl 0607.31001)].