Editing Ordinal utility (section)

== Necessary conditions for existence of ordinal utility function ==
Some conditions on <math>\preceq</math> are necessary to guarantee the existence of a representing function:

* [[Transitive relation|Transitivity]]: if <math>A \preceq B</math> and <math>B \preceq C</math> then <math>A \preceq C</math>. 
* Completeness:   for all bundles <math>A,B\in X</math>: either <math>A\preceq B</math> or <math>B\preceq A</math> or both.
** Completeness also implies reflexivity: for every <math>A\in X</math>: <math>A \preceq A</math>.

When these conditions are met and the set <math>X</math> is finite, it is easy to create a function <math>u</math> which represents <math>\prec</math> by just assigning an appropriate number to each element of  <math>X</math>, as exemplified in the opening paragraph. The same is true when X is [[countably infinite]]. Moreover, it is possible to inductively construct a representing utility function whose values are in the range <math>(-1,1)</math>.<ref name=Rubinstein>Ariel Rubinstein, Lecture Notes in Microeconomic Theory, [http://press.princeton.edu/rubinstein/lecture2.pdf Lecture 2 – Utility]</ref>

When <math>X</math> is infinite, these conditions are insufficient. For example, [[lexicographic preferences]] are transitive and complete, but they cannot be represented by any utility function.<ref name=Rubinstein/> The additional condition required is [[#Continuity|continuity]].