From Wikipedia, the free encyclopedia
Terms in Maths
In
mathematics , a
function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
is said to be closed if for each
α
∈
R
{\displaystyle \alpha \in \mathbb {R} }
, the
sublevel set
{
x
∈
dom
f
|
f
(
x
)
≤
α
}
{\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}}
is a
closed set .
Equivalently, if the
epigraph defined by
epi
f
=
{
(
x
,
t
)
∈
R
n
+
1
|
x
∈
dom
f
,
f
(
x
)
≤
t
}
{\displaystyle {\mbox{epi}}f=\{(x,t)\in \mathbb {R} ^{n+1}\vert x\in {\mbox{dom}}f,\;f(x)\leq t\}}
is closed, then the function
f
{\displaystyle f}
is closed.
This definition is valid for any function, but most used for
convex functions . A
proper convex function is closed
if and only if it is
lower semi-continuous .
[1]
Properties
If
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
is a
continuous function and
dom
f
{\displaystyle {\mbox{dom}}f}
is closed, then
f
{\displaystyle f}
is closed.
If
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
is a
continuous function and
dom
f
{\displaystyle {\mbox{dom}}f}
is open, then
f
{\displaystyle f}
is closed
if and only if it converges to
∞
{\displaystyle \infty }
along every sequence converging to a
boundary point of
dom
f
{\displaystyle {\mbox{dom}}f}
.
[2]
A closed proper convex function f is the pointwise
supremum of the collection of all
affine functions h such that h ≤ f (called the affine minorants of f ).
References