Baire classification of fragmented maps and approximation of separately continuous functions

Abstract

We study a class of fragmented maps which contains all barely continuous maps, scatteredly continuous maps and maps which are pointwise discontinuous on each closed set. We prove that for any Hausdorff space X, a metric space Z and a fragmented map \(f:X\rightarrow Z\) there exists a pointwisely convergent to f sequence of continuous functions \(f_n:X\rightarrow Z\) with an additional condition (which is a weakening of the uniform convergence at each point) in the following cases: (a) X is stratifiable and Z is locally convex equiconnected; (b) X is stratifiable with \(\dim X<\infty \) and Z is equiconnected; (c) X is stratifiable with \(\dim X=0\). We show that for a hereditarily Baire space X, a compact space Y and a metric space Z every separately continuous map Open image in new window is a pointwise limit of a sequence of continuous maps which is uniformly convergent to f with respect to each variable at every point of Open image in new window in cases (a)–(c); (d) X is a perfect zero-dimensional compact space; and (e) X is the Sorgenfrey line.