Suppose f is a function from a
topological space X into a topological space Y. Many classical theorems of topology
assert that from continuity of f follow certain other properties. For example, if f is
continuous, then its point inverses must be closed (if Y is a T1 space) and compact
subsets of X must have compact images under f, that is, f must be compact
preserving. Also, f must be connected, that is, connected subsets of X have
connected images. It is natural to ask whether some combination of these properties
is equivalent to continuity. In particular, our main result (Theorem 2) concerns
compact preserving, connected functions. We give a proof of a result announced but
inadequately proved by E. Halfar that such a function f is continuous if X is a
locally connected Hausdorff space with property K and Y is a Hausdorff space. (A
space X has property K if given that x0 is a limit point of an infinite set A ⊂ X,
then there is a compact set K ⊂ A ∪{x0} such that x0 is a limit point of
K.)
