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 $2^{\aleph_0}=\beth_1=\frak{c}$.