You mean O(1) (alternatively, O(k)) -- from the definition of Big O notation, O(0) is nonsensical. But even then; O(1) just means "constant time", it does not mean "soon."
> from the definition of Big O notation, O(0) is nonsensical
Nitpick: not sure it's nonsensical. Plug g(x) = 0 into the usual definition, and you get |f(x)| ≤ k⋅0 for some k in R, which reduces to f(x) = 0. Which is not satisfiable by any nontrivial algorithm, so not very useful, but not nonsensical.