Slowly Oscillating Continuity

Abstract EN

A function f is continuous if and only if, for each point x0 in the domain, limn→∞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 limλ→1+lim―nmaxn+1≤k≤[λn]|xk-xn|=0, 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.