Let $\Bbb{R}^{+}\\_{0}$ be the set of non-negative real numbers and $\Bbb{R}^{+}$be the set of positive reals. Let us say that a function $f \colon \Bbb{R}^{+}\\_{0} \to \Bbb{R}^{+}\\_{0}$ is **eventually sublinear** if $\ \forall r \in \Bbb{R}^{+} \ \exists x\_0 \in \Bbb{R}^{+} \colon \forall x \geq x\_0, f(x) < rx$. Let $S$ be the set of non-decreasing, eventually sublinear functions (we require no continuity). We define an equivalence relation on $S$: $f \sim g$ if and only if $f$ and $g$ are eventually boundedly close, i.e., $\exists K>0\ \exists x\_0 \in \Bbb{R}^{+} \colon \forall x \geq x\_0, |f(x)-g(x)| < K$. Denote the set of equivalence classes of $S$ under $\sim$ by $\ T$. 


Question 1: What is the cardinality of $\ T$?


Question 2: Does this cardinality change if we drop the requirement that the functions are non-decreasing? What if the co-domain becomes $\Bbb{R}$?


Looking at the set of pairwise non-equivalent sublinear functions $\{f(x) = r\*ln(x) \mid r \in \Bbb{R}\}$ shows that $|T|$ must be at least as big as the cardinality of the reals, $\beth\_1$. Also, $|T|$ can be no bigger than the cardinality of *all* functions from $\Bbb{R}$ to $\Bbb{R}$, which is $\beth\_2$, the cardinality of the power set of the reals. So there seem to be three options:


Option 1: ZFC proves $|T| = \beth\_1$. 


Option 2: ZFC proves $|T| = \beth\_2$.


Option 3: This question is independent from ZFC (but perhaps not independent from ZFC + Generalized CH?).


Different proof ideas involving explicit bounds or forcing are failing me. So does anyone know of an answer and a proof? Thanks. 



Cardinality of Equivalence Relation of Eventually Sublinear Functions
There are only $\beth\_1$ non-decreasing functions from $\mathbb R\_0^+$ to itself, because such a function is determined by its values on a dense set plus information about its countably many discontinuities. So the answer to Question 1 is $\beth\_1$. For Question 2, the answer is $\beth\_2$. Take any proper nonempty subset $A$ of the unit interval (there are $\beth\_2$ of these), and consider the function $f\_A$ that agrees with the natural logarithm at points whose fractional part is in $A$ and is identically 0 at all other points. No two of these functions are boundedly close.


