I am aware of the big theta notation $f = \Theta(g)$ if and only if there are positive constants $A, B$ and $x_0 > 0$ such that for all $x > x_0$ we have $ A|g(x)| \leq |f(x)| \leq B |g(x)|. $ What if the condition is the following: $ C_1 + A|g(x)| \leq |f(x)| \leq C_2 + B |g(x)| $ where $C_1, C_2$ are possibly negative? Certainly more can be said than just $f = O(g)$. Is there a generalized $\Theta$ notation which allows shifts (by, say $C_1, C_2$)? In particular, I'm interested in the special case: \begin{eqnarray} -C \leq f(x) - g(x) \leq C \end{eqnarray} for some positive $C$. How does $f$ compare to $g$ in this case? If $f$ and $g$ are positive functions of $x$ which both diverge to $\infty$, is it true that $f(x) = -C + g(x) + \Theta(1)$? What is the appropriate asymptotic notation in this case?
Update Thanks for the clarifying answers. Now here is a slightly harder question. Suppose $f$ is discrete and $g$ is continuous. Suppose further that as $x \to \infty$, the difference $f(x) - g(x)$ is asymptotically bounded in the interval $[-C,C]$ but does not necessarily converge to $0$. Does $f \sim g$ still make sense? Would it be more appropriate to use $\liminf_{x \to \infty} f(x) - g(x) = - C$ and $\limsup_{x \to \infty} f(x) - g(x) = C$?