Slowly Oscillating Continuity

A function f is continuous if and only if, for each point x0 in the domain, lim⁡n→∞f(xn) = f(x0), whenever lim⁡n→∞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 , where [λn] denotes the integer part of λn. Using ɛ > 0′s and δ′s, this is equivalent to the case when, for any given ɛ > 0, there exist δ = δ(ɛ) > 0 and N = N(ɛ) such that |xm − xn| < ɛ if n ≥ N(ɛ) and n ≤ m ≤ (1 + δ)n. A new type compactness is also defined and some new results related to compactness are obtained.