Editing Ordinal utility (section)

== Continuity ==
A preference relation is called ''continuous'' if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions:

# For every <math>A\in X</math>, the set <math>\{(A,B)|A\preceq B\}</math> is [[topologically closed]] in <math>X\times X</math> with the [[product topology]] (this definition requires <math>X</math> to be a [[topological space]]).
# For every sequence <math>(A_i,B_i)</math>, if for all ''i'' <math>A_i\preceq B_i</math> and <math>A_i \to A</math> and <math>B_i \to B</math>, then <math>A\preceq B</math>.
# For every <math>A,B\in X</math> such that <math>A\prec B</math>, there exists a ball around <math>A</math> and a ball around <math>B</math> such that, for every <math>a</math> in the ball around <math>A</math> and every <math>b</math> in the ball around <math>B</math>, <math>a\prec b</math> (this definition requires <math>X</math> to be a [[metric space]]).

If a preference relation is represented by a continuous utility function, then it is clearly continuous. By the theorems of [[Debreu theorems|Debreu (1954)]], the opposite is also true: 
::Every continuous complete preference relation can be represented by a continuous ordinal utility function.

Note that the [[lexicographic preferences]] are not continuous. For example, <math>(5,0)\prec (5,1)</math>, but in every ball around (5,1) there are points with <math>x<5</math> and these points are inferior to <math>(5,0)</math>. This is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function.