In Debreu's 1954 ordinal utility representation theorem, the utility is unique up to a positive monotonic transformation.
While the uniqueness result is well-known, I fail to find a proper reference. Debreu's original paper does not contain that uniqueness result.
Is there a good reference? Who was the first researcher find this uniqueness result?
The other answer explained why the result is trivial; here is why it is not true without the modifier "on the range" and under the most literal reading.
Consider $[0,1]\cup(2,3]$ with the usual order. Consider the utility representations $u_1$ and $u_2$ with $u_1$ given by $u_1(x)=x$ for all $x$ and $u_2(x)=x$ for $x\leq 2$ and $u_2(x)=x-1$ for $x>2$. There is no strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $u_2=f\circ u_1$. Indeed, that would require $f(3/2)>1$ and $f(3/2)<1+\epsilon$ for all $\epsilon>0$, which is impossible.
It is not a 'result' because it is trivial to show that a positive strictly monotonic transformation function applied to the function that represents the preferences leaves the ordering unchanged. Say $f(x)$ represents a set of preferences on $X$, in the sense that $x \succeq y$ if and only if $f(x) \geq f(y)$. Then let $g(\cdot)$ represent a strictly positive monotonic transformation on the range of $f$. That also means that $g(a) \geq g(b)$ if and only if $a \geq b$. Then it must be true that $x \succeq y$ if and only if $g(f(x)) \geq g(f(y))$. I'll let you fill in the rest of the details.
What that means is that you can create equivalence classes of functions that represent the same ordering. This is what is meant by 'unique up to a positive monotonic transformation'.
