A function f from a topological space X to a topological space Y is called CCCC if the image of every compact subset of X is closed and countably compact in Y. We prove that if X is sequential, then every CCCC function with closed fibers is continuous. We show that every CCCC function f : X → Y where X is locally connected, Frechet, and Y is T2, that is also connectedness-preserving is continuous. Several continuity conditions for CCCC functions that are also connectedness-preserving are obtained. We also prove that every CCCC function on a strong Frechet space is compactness-preserving. Finally it is proved that for a Frechet space X and a T¬2 space Y, if f : X → Y is CCCC then the restriction f | SIf is continuous where SIf of all points x  X at which f is sequentially infinite. MSC: 54C05; 54D05; 54D30