Slowly oscillating continuity
A function f is continuous if and only if, for each point x(0) in the domain, lim(n ->infinity) f (x(n))= f (x(0)), whenever lim(n ->infinity) 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 lim(lambda -> 1+)(lim) over bar (n)max(n+1 <= k <=[lambda n])vertical bar x(k)-x(n)vertical bar = 0, where [lambda n] denotes the integer part of lambda n. Using epsilon > 0's and delta's, this is equivalent to the case when, for any given e > 0, there exist delta = delta(epsilon) > 0 and N = N(epsilon) such that vertical bar x(m) - x(n)vertical bar < epsilon if n >= N(epsilon) and n <= m <= (1 + delta)n. A new type compactness is also defined and some new results related to compactness are obtained. Copyright (c) 2008 H. Cakalli.