Let $\succeq$ be a relation on a set $X$. The function $u: X\to \mathbb{R}$ represents the relation $\succeq$ if $x\succeq y \iff u(x)\geq u(y)$.
I am looking for a good reference on questions such as:
- What conditions on $\succeq$ guarantee that it can be represented by a function?
- What conditions guarantee that it can be represented by a certain kind of function, e.g, continuous, additive, etc.?
- When is the representing function unique (up to a certain transformation)?
These questions have been studied in economics, e.g, for representing an ordinal utility relation by a cardinal utility function. But since they are general mathematics questions, it is possible that they have also been studied in mathematics.
I found some papers presenting specific results of the kind I described above, but I am looking for a referene which summarizes the results in an orderly manner.