Insertion of a function belonging to a certain subclass of RX

Abstract. Let X be a topological space and E(X, R) be a subset of R X with the following properties: (1) Any constant function is in E(X, R); (2) If α, β ∈ R and f, g ∈ E(X, R), then αf + βg ∈ E(X, R); (3) If (f n ) is a sequence of functions in E(X, R) and (f n ) is uniformly convergent to f , then f ∈ E(X, R); and (4) If f ∈ E(X, R) and g is a constant function, then sup{f, g} ∈ E(X, R) and inf{f, g} ∈ E(X, R). Here, necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a function of E(X, R) between two comparable real-valued functions with a certain pair of a general class of properties