φ-statistically quasi Cauchy sequences

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 . Let ( φn ) be a non-decreasing sequence of positive integers such that
nφn +1 ≤ (n + 1) φn for all n ∈ N and the class of all sequences ( φn ) is denoted by . Let E ⊆ N . The
number δφ (E) = lim s →∞
1
φs
|{ k ∈ σ, σ ∈ P s : k ∈ E}| is said to be the φ-density of E . A sequence ( x n )
of points in R is φ-statistically convergent (or S φ-convergent) to a real number  for every ε > 0 if
the set { n ∈ N : | x n −  | ≥ ε} has φ-density zero. We introduce φ-statistically ward continuity of a real
function. A real function is φ-statistically ward continuous if it preserves φ-statistically quasi Cauchy
sequences where a sequence ( x n ) is called to be φ-statistically quasi Cauchy (or S φ-quasi Cauchy)
when (x n ) = (x n +1 − x n ) is φ-statistically convergent to 0. i.e. a sequence ( x n ) of points in R is
called φ-statistically quasi Cauchy (or S φ-quasi Cauchy) for every ε > 0 if { n ∈ N : | x n +1 − x n | ≥ ε}
has φ-density zero. Also we introduce the concept of φ-statistically ward compactness and obtain
results related to φ-statistically ward continuity, φ-statistically ward compactness, statistically ward
continuity, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary
continuity, δ-ward continuity, and slowly oscillating continuity.