Since every continuous map is compact-preserving, the question naturally arises, under what additional conditions a compact-preserving map becomes continuous. In [Proc. Am. Math. Soc. 11, 688-691 (1960; Zbl 0096.17002)] \textit{E. Halfar} investigated this problem for surjections between \(T_2\)-spaces. He proved that if \(X\) is locally compact then every compact-preserving map \(f:X\to Y\), which has closed fibers, is continuous. In [Pac. J. Math. 25, 495-509 (1968; Zbl 0165.25304)] \textit{R. V. Fuller} generalized Halfar's results. In [ibid. 59, 505-514 (1975; Zbl 0324.54007)] \textit{N. Liden} obtained the following generalization of Fuller's result: If \(X\) is a \(T_2\) \(k\)-space and \(Y\) is a \(KC\) space, then a surjection \(f:X\to Y\) is continuous iff \(f\) is compact-preserving and has closed fibers. In this paper the authors give analogues of Liden's theorem and obtain characterizations of perfect maps in terms of compact and compact-preserving maps. Examples are given to show that none of the conditions on the domain and range spaces in their theorems can be dropped. Simple alternatives to two complicated examples of Fuller [loc. cit.] and Halfar [loc. cit.] are also provided.