Multivalued function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a <a href="/facts/Set-valued_function/WH5ua6BL">set-valued function</a> with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions, but English Wikipedia currently does, having a separate article for each. 
A multivalued function of sets f : X → Y is a subset

Γ
          
            f
          
        
         
        ⊆
         
        X
        ×
        Y
        .
      
    
    {\displaystyle \Gamma _{f}\ \subseteq \ X\times Y.}

Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its <a href="/facts/Graph_of_a_function/htYX0BGX">graph</a>

Γ
          
            f
          
        
         
        =
         
        {
        (
        x
        ,
        f
        (
        x
        )
        )
         
        :
         
        x
        ∈
        X
        }
        .
      
    
    {\displaystyle \Gamma _{f}\ =\ \{(x,f(x))\ :\ x\in X\}.}

They are called single-valued functions to distinguish them.

Multivalued function open-in-new

Multivalued function