We introduce lower and upper semi-continuity of a map to the Banach space c(0)(lambda) for an infinite cardinal lambda. We prove that the following conditions (i), (ii) and (iii) on a T-1-space X are equivalent: (i) For every two maps g, h : X --> c(0)(lambda) such that g is upper semi-continuous, h is lower semi-continuous and g less than or equal to h, there exists a continuous map f : X --> c(0)(lambda), with g less than or equal to f less than or equal to h. (ii) For every Banach space Y, with w(Y) less than or equal to lambda, every lower semi-continuous set-valued mapping phi : X --> C-c(Y) admits a continuous selection, where C-c(Y) is the set of all non-empty compact convex sets in Y. (iii) X is normal and every locally finite family F of subsets of X with \F\ < lambda, has a locally finite Open expansion provided it has a point-finite open expansion. We also characterize several paracompact-like properties by inserting continuous maps between semi-continuous Banach-valued functions.