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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 (@Dx"n) is a null sequence. In this paper we call a real-valued function defined on a subset E of [email protected] continuous if it preserves @d-quasi-Cauchy sequences where a sequence x=(x"n) is defined to be @d-quasi-Cauchy if the sequence (@Dx"n) is quasi-Cauchy. It turns out that @d-ward continuity implies uniform continuity, but there are uniformly continuous functions which are not @d-ward continuous. A new type of compactness in terms of @d-quasi-Cauchy sequences, namely @d-ward compactness is also introduced, and some theorems related to @d-ward continuity and @d-ward compactness are obtained.