which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.There is an indicator function for affine varieties over a finite field: given a finite set of functions
f
α
∈
F
q
[
x
1
,
…
,
x
n
]
{\displaystyle f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]}
let
V
=
{
x
∈
F
q
n
:
f
α
(
x
)
=
0
}
{\displaystyle V=\left\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\right\}}
be their vanishing locus. Then, the function
P
(
x
)
=
∏
(
1
−
f
α
(
x
)
q
−
1
)
{\textstyle P(x)=\prod \left(1-f_{\alpha }(x)^{q-1}\right)}
acts as an indicator function for
V
{\displaystyle V}
. If
x
∈
V
{\displaystyle x\in V}
then
P
(
x
)
=
1
{\displaystyle P(x)=1}
, otherwise, for some
f
α
{\displaystyle f_{\alpha }}
, we have
f
α
(
x
)
≠
0
{\displaystyle f_{\alpha }(x)\neq 0}
, which implies that
f
α
(
x
)
q
−
1
=
1
{\displaystyle f_{\alpha }(x)^{q-1}=1}
, hence
P
(
x
)
=
0
{\displaystyle P(x)=0}
.
The characteristic function in convex analysis, closely related to the indicator function of a set:
In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:
{\displaystyle \operatorname {E} }
denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.
The characteristic function of a cooperative game in game theory.
The characteristic polynomial in linear algebra.
The characteristic state function in statistical mechanics.
The Euler characteristic, a topological invariant.
The receiver operating characteristic in statistical decision theory.
The point characteristic function in statistics.