I\'m trying to determine a function f ( x ) that is O ( f ) but not o ( f ) and
ID: 647069 • Letter: I
Question
I'm trying to determine a function f(x) that is O(f) but not o(f) and also not ?(f). Note the f used in the asymptotic notation is not the same as f(x).
Originally I thought of f(x)=log(x),O(x) but I am not convinced that o(x) is invalid for this function.
Previously I thought it was, because I could always come up with some constant c that would bring the function x below f(x). However, I could say the same for o(2x) because surely there is some infinitesimally small constant that I can find that will put 2x below f(x) at a given value x. Any advice in this matter?
Explanation / Answer
Your example doesn't work, since in fact logx=o(x). However, the function ? below is O(x) but not o(x) or ?(x):
?(x)={x1if ?x? is even,if ?x? is odd.}