(UTC) 

 My interpretation is that O(g(x)) denotes an unspecified (or anonymous) function that is asymptotically bounded by a multiple of g, rather than the class of all such functions. This is similar to how we use the constant of integration (+C) in calculus. We say that the most general antiderivative of cos(x) is sin(x)+C, but almost nobody says that C denotes the set of all constant functions  rather, it is an "arbitrary constant". So what is wrong with saying that O(1) is an "arbitrary bounded function" and writing sin(x) = O(1)? 22:24, 17 March 2008