Separating sets by peripherally continuous functions (Q2810999)

A function \(f:\mathbb R\to \mathbb R\) is peripherally continuous if for each \(x\in \mathbb R\) there are sequences \((a_n)\) and \((b_n)\) such that \(a_n\to x^+\), \(b_n\to x^-\), \(f(a_n)\to f(x)\) and \(f(b_n)\to f(x)\). In this paper, the pairs of disjoint sets \((A_0, A_1)\) which can be separated by peripherally continuous functions (in the sense that there is a peripherally continuous function \(f\) such that \(f=0\) on \(A_0\) and \(f=1\) on \(A_1\)) are characterized. Moreover, the pairs \((A^+, A^-)\) for which there is a peripherally continuous function \(f\) such that \(f>0\) on \(A^+\) and \(f<0\) on \(A^-\) are characterized.