preference relations and monotonic transformations of utillity functions Given a choice set $X$ (NOT assumed to be a commodity set...), and utility functions $u,u'$ on $X$, it is clear that if $u'$ is a strictly monotonic transformation of $u$ then they induce the same preference relation on $X$.

My question is, under what conditions does the opposite hold? What are the least strict conditions we can think of that imply that if $u,u'$ induce the same preference relation on $X$, that $u'$ must then necessarily be a strictly increasing transformation of $u$?
Depending onhow one reads the question, the converse holds either always or basically never. Let $u:X\to\mathbb{R}$ and $v:X\to\mathbb{R}$ be arbitrary functions with ranges $u(X)$ and $v(X)$, respectively. It is not hard to show that $u$ and $v$ represent the same preferences if and only if there exists a strictly increasing function $h:v(X)\to\mathbb{R}$ such that $u=h\circ v$. So this is the "always"-answer.

Now one might require $h$ to be defined and strictly increasing on all of $\mathbb{R}$. In that case, one can basically never guarantee the existence of such an $h$, at least for standard economic applications. Let $X=[0,1]$, $u$ be given by $u(x)=x$ and $v$ be given by $v(x)=x$ for $x\leq 1/2$ and $v(x)=x+1$ for $x&gt;1/2$. Then $v(X)=[0, 1/2]\cup (3/2,2]$. If $h:\mathbb{R}\to\mathbb{R}$ is a nondecreasing function such that $u=h\circ v$, all $r\in(1/2,3/2]$ must be mapped to the same point by $h$, so $h$ cannot be strictly increasing. 

Such a problem occurs whenever $X$ is a connected topological space (say, a convex subset of some $\mathbb{R}^n$) and some continuous utility representation exists that is not constant.

The problem can even occur if both $v$ and $u$ are continuous. Let $X=\mathbb{R}$, $u$ be given by $u(x)=x$ and $v$ be given by $v(x)=e^x$. Then $v(X)=\mathbb{R}_{++}$, the set of strictly positive numbers. Suppose there were some strictly increasing function $h:\mathbb{R}\to\mathbb{R}$ such that $u=h\circ v$. Then $h$ would be a strictly increasing extension of the logarithm function to all of $\mathbb{R}$, which is impossible.

Finally, some positive result: If $X$ is a connected topological space, and $u:X\to\mathbb{R}$ and $v:X\to\mathbb{R}$ are continuous functions that represent the same preferences and are either both bounded above or unbounded above and are either both bounded below or unbounded below, then there exists a strictly increasing continuous function $h:\mathbb{R}\to\mathbb{R}$ such that $u=h\circ v$.