ON phi-IDEAL WARD CONTINUITY

Abstract

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. Let P denote the space whose elements are finite sets of distinct positive integers. Given any element sigma of P; we denote by p (sigma) the sequence {p(n) (sigma) g such that p(n) (sigma) = 1 for n is an element of sigma and p(n) (sigma) = 0 otherwise. Further P-s = {sigma is an element of P : Sigma(infinity)(n-1) p(n) (sigma) <= s}, i.e.P-s is the set of those sigma whose support has cardinality at most s; and Phi = {phi = (phi(n)) : 0 < phi(1) <= phi(n) <= phi(n+1) and n phi(n+1) <= (n + 1)phi(n) }, A sequence (x(n)) of points in R is called phi-ideal convergent ( or I-phi-convergent) to a real number l if for every epsilon > 0

{s is an element of N : 1/phi(s) Sigma(n is an element of sigma,sigma is an element of Ps) vertical bar x(n) - l vertical bar >= epsilon} is an element of I.

We introduce phi-ideal ward continuity of a real function. A real function is phi-ideal ward continuous if it preserves phi-ideal quasi Cauchy sequences where a sequence (x(n)) is called to be phi-ideal quasi Cauchy (or I-phi-quasi Cauchy) when (Delta x(n)) = (x(n+1) - x(n)) is phi-ideal convergent to 0. i. e. a sequence (x(n)) of points in R is called phi-ideal quasi Cauchy (or I-phi-quasi Cauchy) for every epsilon > 0 if

{s is an element of N : 1/phi(s) Sigma(n is an element of sigma,sigma is an element of Ps) vertical bar x(n+1) - x(n)vertical bar >= epsilon} is an element of I.

In this paper, we prove that any phi-ideal continuous function is uniformly continuous either on an interval or on a phi-ideal ward compact subset of R : We also characterize the uniform continuity via phi-ideal quasi-Cauchy sequences.