Some Technical Details

The ``big'' and ``little'' functions behave almost exactly like their standard real-number counterparts: transitivity, reflexivity, symmetry, and transpose symmetry.

However, one standard property does not hold: trichotomy. There are pairs of functions f(n) and g(n) such that none of f(n) = o(g(n)), $f(n) = \Theta(g(n))$, nor $f(n) = \omega(g(n))$ holds. The book gives an example of $n^{1+\sin n}$ which can't be compared directly to n because it often acts like n².