We can easily prove there are $c:=\beth_1$ computational complexity classes for positive-valued functions of one positive-integer variable. To prove there are at least $c$ classes, we note each $e^{an},\,e^{bn}$ with $a>b>0$ are in different complexity classes. To prove there are at most $c$ classes, we note any function $f$'s class is determined by its $\aleph_0$ values on $\mathbb{N}$, giving an upper bound of $c^{\aleph_0}=c$.
But does there exist a map from $\mathbb{R}$ to functions, say $F(x)=n\mapsto f(n)$, such that $x<y\implies F(x)\in o(F(y))$, with the image of $F$ including representatives of all complexity classes?