Fun fact: there exist pairs of positive monotonic functions f(x) and g(x) such that neither f(x)=O(g(x)) nor g(x)=O(f(x)), i.e. the functions are not comparable

Fun fact: there exist pairs of positive monotonic functions f(x) and g(x) such that neither f(x)=O(g(x)) nor g(x)=O(f(x)), i.e. the functions are not comparable asymptotically. Example: f(x)=gamma(floor(x)+1) (=floor(x)!) g(x)=gamma(floor(x-0.5)+1.5) There exists no such constant M>0 that f(x)<M*g(x) for x from some point on, and the converse. The two functions periodically cross each other in a way that neither is asymptotically bound by the other.