Slowly Oscillating Continuity

Çakalli, H.
January 2008
Abstract & Applied Analysis;2008, p1
Academic Journal
A function f is continuous if and only if, for each point x0 in the domain, lim n→∞f(xn) = f (x0), whenever limn→∞xn = x0. This is equivalent to the statement that (f(xn)) is a convergent sequence whenever (xn) is convergent. The concept of slowly oscillating continuity is defined in the sense that a function f is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, (f(xn)) is slowly oscillating whenever (xn) is slowly oscillating. A sequence (xn) of points in R is slowly oscillating if lim ℷ→1+ lim nmaxn+1≤k≤[ℷn] ∣xk - xn∣ = 0, where [ℷn] denotes the integer part of ℷn. Using ϵ > 0's and ∂'s, this is equivalent to the case when, for any given ϵ > 0, there exist ∂ = ∂(ϵ) > 0 and N = N(ϵ) such that ∣xm - xn∣ < ϵ if n ≥ N (ϵ) and n ≤ m ≤ (1 + ∂)n. A new type compactness is also defined and some new results related to compactness are obtained.


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