ON φ-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 σ of P, we denote by p(σ) the sequence {p_{n}(σ)} such that p_{n}(σ)=1 for n∈σ and p_{n}(σ)=0 otherwise. Further P_{s}={σ∈P:∑_{n=1}^{∞}p_{n}(σ)≤s}, i.e. P_{s} is the set of those σ whose support has cardinality at most s, and we get Φ={φ=(φ_{n}):0<φ₁≤φ_{n}≤φ_{n+1}[mbox]<LaTeX>\mbox{~and~}</LaTeX>nφ_{n+1}≤(n+1)φ_{n}}. A sequence (x_{n}) of points in R is called φ-ideal convergent (or I_{φ}-convergent) to a real number ℓ if for every ε>0  {s∈N:(1/(φ_{s}))∑_{n∈σ,σ∈P_{s}}|x_{n}-ℓ|≥ε}∈I. We introduce φ-ideal ward continuity of a real function. A real function is φ-ideal ward continuous if it preserves φ-ideal quasi Cauchy sequences where a sequence (x_{n}) is called to be φ-ideal quasi Cauchy (or I_{φ}-quasi Cauchy) when (Δx_{n})=(x_{n+1}-x_{n}) is φ-ideal convergent to 0. i.e. a sequence (x_{n}) of points in R is called φ-ideal quasi Cauchy (or I_{φ}-quasi Cauchy) for every ε>0 if  {s∈N:(1/(φ_{s}))∑_{n∈σ,σ∈P_{s}}|x_{n+1}-x_{n}|≥ε}∈I. In this paper, we prove that any φ-ideal continuous function is uniformly continuous either on an interval or on a φ-ideal ward compact subset of R. Also we characterize the uniform continuity via φ-ideal quasi-Cauchy sequences.