A study of Foran's conditions \(A(N)\) and \(B(N)\) and his class \({\mathcal F}\) (Q1820882)

For a positive integer \(N\), the conditions \(A(N)\) and \(B(N)\) generalizing absolute continuity and bounded variation, respectively, along with class \({\mathcal F}\), were introduced by \textit{J. Foran} [Proc. Am. Math. Soc. 49, 359--365 (1975; Zbl 0305.26004)]. Main results of the paper under review: 1. For every \(N\), there is a perfect set \(P\) and a continuous map \(f\) on \(P\) such that \(f\in A(N+1)\) on \(P\) but \(f\not\in A(N)\) on every portion of \(P\). 2. \({\mathcal F}\) contains a map which is not in \(B(2)\setminus A(N)\) on the Cantor set, for every \(N\). 3. There is \(f\in {\mathcal F}\) not a.e. approximately differentiable. 4. A new class of maps containing \({\mathcal F}\) is considered.