The family of regular closed subsets of a topological space is
used to introduce two concepts concerning a function f from a
space X to a space Y. The first of them is the notion of f
being rc-continuous. One of the established results states that a
space Y is extremally disconnected if and only if each
continuous function from a space X to Y is rc-continuous. The
second concept studied is the notion of a function f having an
rc-strongly closed graph. Also one of the established results
characterizes rc-compact spaces (≡S-closed spaces) in
terms of functions that possess rc-strongly closed graph.