The title pretty much says it all. What is the cardinality of $G$, the group of all functions $f: \mathbb{R} \to \mathbb{R}$ such that $\forall x,y\in \mathbb{R} \left( x>y\Rightarrow f(x)>f(y)\right)$? Obviously $G$ is a subgroup of the symmetric group on $\mathbb{R}$, which has cardinality $\beth\_2$, and $G$ is also obviously uncountable, so the real question here is whether the cardinality is $\beth\_1$ or $\beth\_2$.



Cardinality of group of order-preserving functions from R to R
Such a function can have only countably many points of discontinuity, since any discontinuity will be a jump discontinuity and hence the range will skip over an interval unique to that point, and so we can associate each point of discontinuity with a distinct rational number, meaning there are only countably many. 


So the function is determined by specifying its values at the members of this countable set, plus specifying values on a countable dense set. This is altogether a countable amount of information, and so the number of such functions is $2^{\aleph\_0}=\beth\_1=\frak{c}$. 


