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Definition

Let X {\displaystyle X} be a set, and let A {\displaystyle A} be a subset of X {\displaystyle X} . The characteristic function of A {\displaystyle A} is the function

χ A : X → R ∪ { + ∞ } {\displaystyle \chi _{A}:X\to \mathbb {R} \cup \{+\infty \}}

taking values in the extended real number line defined by

χ A ( x ) := { 0 , x ∈ A ; + ∞ , x ∉ A . {\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}

Relationship with the indicator function

Let 1 A : X → R {\displaystyle \mathbf {1} _{A}:X\to \mathbb {R} } denote the usual indicator function:

1 A ( x ) := { 1 , x ∈ A ; 0 , x ∉ A . {\displaystyle \mathbf {1} _{A}(x):={\begin{cases}1,&x\in A;\\0,&x\not \in A.\end{cases}}}

If one adopts the conventions that

• for any a ∈ R ∪ { + ∞ } {\displaystyle a\in \mathbb {R} \cup \{+\infty \}} , a + ( + ∞ ) = + ∞ {\displaystyle a+(+\infty )=+\infty } and a ( + ∞ ) = + ∞ {\displaystyle a(+\infty )=+\infty } , except 0 ( + ∞ ) = 0 {\displaystyle 0(+\infty )=0} ;
• 1 0 = + ∞ {\displaystyle {\frac {1}{0}}=+\infty } ; and
• 1 + ∞ = 0 {\displaystyle {\frac {1}{+\infty }}=0} ;

then the indicator and characteristic functions are related by the equations

1 A ( x ) = 1 1 + χ A ( x ) {\displaystyle \mathbf {1} _{A}(x)={\frac {1}{1+\chi _{A}(x)}}}

and

χ A ( x ) = ( + ∞ ) ( 1 − 1 A ( x ) ) . {\displaystyle \chi _{A}(x)=(+\infty )\left(1-\mathbf {1} _{A}(x)\right).}

Subgradient

The subgradient of χ A ( x ) {\displaystyle \chi _{A}(x)} for a set A {\displaystyle A} is the tangent cone of that set in x {\displaystyle x} .

Bibliography

• Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 978-0-691-01586-6.