On variations on quasi Cauchy sequences in metric spaces

For a fixed positive i nteger p, a sequence (xn) in a metric space X is c alled p-quasi-Cauchy if (Δp xn) is a null sequence where Δp xn=d(xn+p, xn) for each positive integer n. A subset E of X is called p-ward compact if any sequence (xn) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f (xn)) is a p-quasi Cauchy sequence in Y whenever (xn) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f (xn) is p-quasi Cauchy in Y whenever (xn) is a quasi cauchy sequence of points in E.