On the class of functions having infinite limit on a given set

According to [1] for a linear set A there exists a function f : R ? R such that A = Lf (R) if and only if A is a countable Gd-set. Our purpose is to prove a similar result in a more general setting and to investigate the cardinality and topological properties of the class of functions f : X ? R for which Lf (X) equals a given non-empty, countable Gd-set.