Recently, it has been proved that a real-valued function defined on a subset E of R, the set of real numbers, is uniformly continuous on E if and only if it is defined on E and preserves quasi-Cauchy sequences of points in E where a sequence is called quasi-Cauchy if (?xn) is a null sequence. In this paper we call a real-valued function defined on a subset E of R?-ward continuous if it preserves ?-quasi-Cauchy sequences where a sequence x=(xn) is defined to be ?-quasi-Cauchy if the sequence (?xn) is quasi-Cauchy. It turns out that ?-ward continuity implies uniform continuity, but there are uniformly continuous functions which are not ?-ward continuous. A new type of compactness in terms of ?-quasi-Cauchy sequences, namely ?-ward compactness is also introduced, and some theorems related to ?-ward continuity and ?-ward compactness are obtained.