little-o

A colleague of mine has asked me a question: Is the set o(1) ( little o notation ) empty? My question is: Is o(1) an empty set? If not, is there a program that has o(1) time complexity? Reminder, definition of little-o by Cormen : A function f(n) is said to be in o(g(n)) if for any positive constant c>0 , there exists a constant n0 > 0 such that 0 <=f(n) < cg(n) , for all n>= n0 . Intuitively, if f(n) is in o(g(n)) if it is in O(g(n)) , but this bound is NOT tight. The set o(1) is not empty . It is first important to remember that f(x) is in o(g(x)) if and only if lim_x->infinity { f(x) / g(x)

What is the difference between Big-O notation O(n) and Little-O notation o(n) ? Tyler McHenry f ∈ O(g) says, essentially For at least one choice of a constant k > 0, you can find a constant a such that the inequality 0 <= f(x) <= k g(x) holds for all x > a. Note that O(g) is the set of all functions for which this condition holds. f ∈ o(g) says, essentially For every choice of a constant k > 0, you can find a constant a such that the inequality 0 <= f(x) < k g(x) holds for all x > a. Once again, note that o(g) is a set. In Big-O, it is only necessary that you find a particular multiplier k for