In economics, a utility representation theorem  asserts that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. 
 Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). If the agent prefers A to B, we write 
  
    
      
        A
        ≻
        B
      
    
    {\displaystyle A\succ B}
  
. The set of all such preference-pairs forms the person's preference relation. 
 Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function - a function u that assigns a real number to each option, such that 
  
    
      
        u
        (
        A
        )
        >
        u
        (
        B
        )
      
    
    {\displaystyle u(A)>u(B)}
  
 if and only if 
  
    
      
        A
        ≻
        B
      
    
    {\displaystyle A\succ B}
  
.
 Not every preference-relation has a utility-function representation. For example, if the relation is not transitive (the agent prefers A to B, B to C, and C to A), then it has no utility representation, since any such utility function would have to satisfy 
  
    
      
        u
        (
        A
        )
        >
        u
        (
        B
        )
        >
        u
        (
        C
        )
        >
        u
        (
        A
        )
      
    
    {\displaystyle u(A)>u(B)>u(C)>u(A)}
  
, which is impossible.
 A utility representation theorem gives conditions on a preference relation, that are sufficient for the existence of a utility representation.
 Often, one would like the representing function u to satisfy additional conditions, such as continuity. This requires additional conditions on the preference relation.
 The set of options is a topological space denoted by X. In some cases we assume that X is also a metric space; in particular, X can be a subset of a Euclidean space Rm, such that each coordinate in {1,...,m} represents a commodity, and each m-vector in X represents a possible consumption bundle.
 A preference relation is a subset of 
  
    
      
        X
        ×
        X
      
    
    {\displaystyle X\times X}
  
. It is denoted by either 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 or 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
:
 Given a weak preference relation 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
, one can define its "strict part" 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 and "indifference part" 
  
    
      
        ≃
      
    
    {\displaystyle \simeq }
  
 as follows:
 Given a strict preference relation 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
, one can define its "weak part" 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 and "indifference part" 
  
    
      
        ≃
      
    
    {\displaystyle \simeq }
  
 as follows:
 For every option 
  
    
      
        A
        ∈
        X
      
    
    {\displaystyle A\in X}
  
, we define the contour sets at A: 
 Sometimes, the above continuity notions are called semicontinuous, and a 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 is called continuous if it is a closed subset of 
  
    
      
        X
        ×
        X
      
    
    {\displaystyle X\times X}
  
.
 A preference-relation is called:
 As an example, the strict order ">" on real numbers is separable, but not countable. 
 A utility function is a function 
  
    
      
        u
        :
        X
        →
        
          R
        
      
    
    {\displaystyle u:X\to \mathbb {R} }
  
.
 Debreu proved the existence of a contiuous representation of a weak preference relation 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 satisfying the following conditions:
 Jaffray gives an elementary proof to the existence of a continuous utility function.
 Preferences are called incomplete when some options are incomparable, that is, neither 
  
    
      
        A
        ⪰
        B
      
    
    {\displaystyle A\succeq B}
  
 nor  
  
    
      
        B
        ⪰
        A
      
    
    {\displaystyle B\succeq A}
  
 holds. This case is denoted by 
  
    
      
        A
        ⋈
        B
      
    
    {\displaystyle A\bowtie B}
  
. Since real numbers are always comparable, it is impossible to have a representing function u with 
  
    
      
        u
        (
        A
        )
        ≥
        u
        (
        B
        )
        
        ⟺
        
        A
        ⪰
        B
      
    
    {\displaystyle u(A)\geq u(B)\iff A\succeq B}
  
. There are several ways to cope with this issue. 
 Peleg defined a utility function representation of a strict partial order 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 as a function 
  
    
      
        u
        :
        X
        →
        
          R
        
      
    
    {\displaystyle u:X\to \mathbb {R} }
  
 such that
  
    
      
        A
        ≻
        B
        
        ⟹
        
        u
        (
        A
        )
        >
        u
        (
        B
        )
      
    
    {\displaystyle A\succ B\implies u(A)>u(B)}
  
, that is, only one direction of implication should hold. Peleg proved the existence of a one-dimensional continuous utility representation of a strict preference relation 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 satisfying the following conditions:   
 If we are given a weak preference relation 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
, we can apply Peleg's theorem by defining a strict preference relation: 
  
    
      
        A
        ≻
        B
      
    
    {\displaystyle A\succ B}
  
 if and only if 
  
    
      
        A
        ⪰
        B
      
    
    {\displaystyle A\succeq B}
  
 and not 
  
    
      
        B
        ⪰
        A
      
    
    {\displaystyle B\succeq A}
  
.
 The second condition (
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 is separable) is implied by the following three conditions:
 A similar approach was taken by Richter. Therefore, this one-directional representation is also called a Richter-Peleg utility representation.
 Jaffray defines a utility function representation of a strict partial order 
  
    
      
        ≻
      
    
    {\displaystyle \succ }
  
 as a function 
  
    
      
        u
        :
        X
        →
        
          R
        
      
    
    {\displaystyle u:X\to \mathbb {R} }
  
 such that both 
  
    
      
        A
        ≻
        B
        
        ⟹
        
        u
        (
        A
        )
        >
        u
        (
        B
        )
      
    
    {\displaystyle A\succ B\implies u(A)>u(B)}
  
, and 
  
    
      
        A
        ≈
        B
        
        ⟹
        
        u
        (
        A
        )
        =
        u
        (
        B
        )
      
    
    {\displaystyle A\approx B\implies u(A)=u(B)}
  
, where the relation 
  
    
      
        A
        ≈
        B
      
    
    {\displaystyle A\approx B}
  
 is defined by: for all C, 
  
    
      
        A
        ≻
        C
        
        ⟺
        
        B
        ≻
        C
      
    
    {\displaystyle A\succ C\iff B\succ C}
  
 and 
  
    
      
        C
        ≻
        A
        
        ⟺
        
        C
        ≻
        B
      
    
    {\displaystyle C\succ A\iff C\succ B}
  
 (that is: the lower and upper contour sets of A and B are identical). He proved that, for every partially-ordered space 
  
    
      
        (
        X
        ,
        ≻
        )
      
    
    {\displaystyle (X,\succ )}
  
 that is perfectly-separable, there exists a utility function that is upper-semicontinuous in any topology stronger than the upper order topology.: Sec.4  An analogous statement states the existence of a utility function that is lower-semicontinuous in any topology stronger than the lower order topology.
 Sondermann defines a utility function representation similarly to Jaffray. He gives conditions for existence of a utility function representation on a probability space, that is upper semicontinuous or lower semicontinuous in the order topology.
 Herden defines a utility function representation of a weak preorder 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 as an isotone function[clarification needed] 
  
    
      
        u
        :
        (
        X
        ,
        ⪰
        )
        →
        (
        
          R
        
        ,
        ≥
        )
      
    
    {\displaystyle u:(X,\succeq )\to (\mathbb {R} ,\geq )}
  
 such that 
  
    
      
        A
        ≻
        B
        
        ⟹
        
        u
        (
        A
        )
        >
        u
        (
        B
        )
      
    
    {\displaystyle A\succ B\implies u(A)>u(B)}
  
. Herden: Thm.4.1  proved that  a weak preorder 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 on X has a continuous utility function, if and only if there exists a countable family E of separable systems on X such that, for all pairs 
  
    
      
        A
        ≻
        B
      
    
    {\displaystyle A\succ B}
  
, there is a separable system F in E, such that B is contained in all sets in F, and A is not contained in any set in F. He shows that this theorem implies Peleg's representation theorem. In a follow-up paper he clarifies the relation between this theorem and classical utility representation theorems on complete orders.
 A multi-utility representation (MUR) of a relation 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 is a set U of utility functions, such that 
  
    
      
        A
        ⪰
        B
        
        ⟺
        
        ∀
        u
        ∈
        U
        :
        u
        (
        A
        )
        ≥
        u
        (
        B
        )
      
    
    {\displaystyle A\succeq B\iff \forall u\in U:u(A)\geq u(B)}
  
. In other words, A is preferred to B if and only if all utility functions in the set U unanimously hold this preference. The concept was introduced by Efe Ok.    
 Every preorder (reflexive and transitive relation) has a trivial MUR.: Prop.1  Moreover, every preorder with closed upper contour sets has an upper-semicontinuous MUR, and every preorder with closed lower contour sets has a lower-semicontinuous MUR.: Prop.2  However, not every preorder with closed upper and lower contour sets has a continuous MUR.: Exm.1  Ok and Evren present several conditions on the existence of a continuous MUR:   
 All the representations guaranteed by the above theorems might contain infinitely many utilities, and even uncountably many utilities. In practice, it is often important to have a finite MUR - a MUR with finitely many utilities. Evren and Ok prove there exists a finite MUR where all utilities are upper[lower] semicontinuous for any weak preference relation 
  
    
      
        ⪰
      
    
    {\displaystyle \succeq }
  
 satisfying the following conditions: : Thm 3 
 Note that the guaranteed functions are semicontinuous, but not necessarily continuous, even if all upper and lower contour sets are closed.: Exm.2   Evren and Ok say that "there does not seem to be a natural way of deriving a continuous finite multi-utility representation theorem, at least, not by using the methods adopted in this paper".