The continuity properties of compact-preserving functions

A function f : X → Y between topological spaces is called compact-preserving if the image f ( K ) of each compact subset K ⊂ X is compact. We prove that a function f : X → Y defined on a strong Fréchet space X is compact-preserving if and only if for each point x ∈ X there is a compact subset K x ⊂ Y such that for each neighborhood O f ( x ) ⊂ Y of f ( x ) there is a neighborhood O x ⊂ X of x such that f ( O x ) ⊂ O f ( x ) ∪ K x and the set K x ∖ O f ( x ) is finite. This characterization is applied to give an alternative proof of a classical characterization of continuous functions on locally connected metrizable spaces as functions that preserve compact and connected sets. Also we show that for each compact-preserving function f : X → Y defined on a (strong) Fréchet space X, the restriction f | L I f ′ (resp. f | L I f ) is continuous. Here L I f is the set of points x ∈ X of local infinity of f and L I f ′ is the set of non-isolated points of the set L I f . Suitable examples show that the obtained results cannot be improved.