GENERALIZED CONTINUITY AND UNIFORM APPROXIMATION BY STEP FUNCTIONS.

Given two topological spaces X and Y and a family 풪* of subsets of X, a function f : X → Y is called 풪*-continuous if f-1 (V ) ∈ 풪* for every open set V ⊆ Y . An 풪*-step function is meant to be a function 훗 : X → Y that is piecewise constant on a partition of X into sets from 풪*. Using some technical assumptions on X, Y , and 풪* we give representations of 풪*-continuous functions as uniform limits of 풪*-step functions. We deal in particular with α-continuous, nearly continuous, almost quasi-continuous, and somewhat continuous functions. The paper is motivated by a corresponding characterization of quasi-continuous functions. [ABSTRACT FROM AUTHOR]