Slowly Oscillating Continuity

A function f is continuous if and only if, for each point x 0 in the domain, lim⁡ n → ∞ f ( x n ) = f ( x 0 ), whenever lim⁡ n → ∞ x n = x 0 . This is equivalent to the statement that ( f ( x n )) is a convergent sequence whenever ( x n ) 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 ( x n )) is slowly oscillating whenever ( x n ) is slowly oscillating. A sequence ( x n ) of points in R is slowly oscillating if , where [ λ n ] denotes the integer part of λ n . 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 | x m − x n | < 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.