Semi-continuity explained

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function

is upper (respectively, lower) semicontinuous at a point

x₀

if, roughly speaking, the function values for arguments near

x₀

are not much higher (respectively, lower) than

f\left(x_0\right).

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point

x₀

f\left(x_0\right)+c

for some

c>0

, then the result is upper semicontinuous; if we decrease its value to

f\left(x_0\right)-c

then the result is lower semicontinuous.

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.^[1]

Definitions

Assume throughout that

is a topological space and

f:X\to\overline{\R}

is a function with values in the extended real numbers

\overline{\R}=\R\cup\{-infty,infty\}=[-infty,infty]

Upper semicontinuity

A function

f:X\to\overline{\R}

is called upper semicontinuous at a point

x₀\inX

if for every real

y>f\left(x_0\right)

there exists a neighborhood

x₀

such that

f(x)<y

for all

x\inU

.^[2] Equivalently,

is upper semicontinuous at

x₀

if and only if

\limsup_ f(x) \leq f(x_0)

where lim sup is the limit superior of the function

at the point

x_0.

is a metric space with distance function

and

f(x_0)\in\R,

this can also be restated using an

\varepsilon

\delta

formulation, similar to the definition of continuous function. Namely, for each

\varepsilon>0

there is a

\delta>0

such that

f(x)<f(x_{0)+\varepsilon}

whenever

d(x,x_0)<\delta.

A function

f:X\to\overline{\R}

is called upper semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is upper semicontinuous at every point of its domain.

(2) For each

y\in\R

, the set

f^-1([-infty,y))=\{x\inX:f(x)<y\}

is open in

, where

[-infty,y)=\{t\in\overline{\R}:t<y\}

(3) For each

y\in\R

, the

-superlevel set

f^-1([y,infty))=\{x\inX:f(x)\gey\}

is closed in

\{(x,t)\inX x \R:t\lef(x)\}

is closed in

X x \R

(5) The function

is continuous when the codomain

\overline{\R}

is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals

[-infty,y)

Lower semicontinuity

A function

f:X\to\overline{\R}

is called lower semicontinuous at a point

x_0\inX

if for every real

y<f\left(x_0\right)

there exists a neighborhood

x₀

such that

f(x)>y

for all

x\inU

.Equivalently,

is lower semicontinuous at

x₀

if and only if

\liminf_ f(x) \ge f(x_0)

where

\liminf

is the limit inferior of the function

at point

x_0.

is a metric space with distance function

and

f(x_0)\in\R,

this can also be restated as follows: For each

\varepsilon>0

there is a

\delta>0

such that

f(x)>f(x_{0)-\varepsilon}

whenever

d(x,x_0)<\delta.

A function

f:X\to\overline{\R}

is called lower semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is lower semicontinuous at every point of its domain.

(2) For each

y\in\R

, the set

f^-1((y,infty])=\{x\inX:f(x)>y\}

is open in

, where

(y,infty]=\{t\in\overline{\R}:t>y\}

(3) For each

y\in\R

, the

-sublevel set

f^-1((-infty,y])=\{x\inX:f(x)\ley\}

is closed in

\{(x,t)\inX x \R:t\gef(x)\}

is closed in

X x \R

.^[3]

(5) The function

is continuous when the codomain

\overline{\R}

is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals

(y,infty]

Examples

Consider the function

piecewise defined by:

f(x) = \begin-1 & \mbox x < 0,\\ 1 & \mbox x \geq 0\end

This function is upper semicontinuous at

x₀=0,

but not lower semicontinuous.

f(x)=\lfloorx\rfloor,

which returns the greatest integer less than or equal to a given real number

is everywhere upper semicontinuous. Similarly, the ceiling function

f(x)=\lceilx\rceil

is lower semicontinuous.

Upper and lower semicontinuity bear no relation to continuity from the left or from the right for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.^[4] For example the function $f(x) = \begin\sin(1/x) & \mbox x \neq 0,\\1 & \mbox x = 0,\end$ is upper semicontinuous at

x=0

while the function limits from the left or right at zero do not even exist.

X=\Rⁿ

is a Euclidean space (or more generally, a metric space) and

\Gamma=C([0,1],X)

is the space of curves in

(with the supremum distance

d_{\Gamma(\alpha,\beta)}=\sup\{d_{X(\alpha(t),\beta(t)):t\in[0,1]\}}

), then the length functional

L:\Gamma\to[0,+infty],

which assigns to each curve

\alpha

its length

L(\alpha),

is lower semicontinuous.^[5] As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length

\sqrt2

Let

(X,\mu)

be a measure space and let

L^+(X,\mu)

denote the set of positive measurable functions endowed with thetopology of convergence in measure with respect to

\mu.

Then by Fatou's lemma the integral, seen as an operator from

L^+(X,\mu)

[-infty,+infty]

is lower semicontinuous.

Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on L^p spaces in terms of the convexity of another function.

Properties

to the extended real numbers

\overline{\R}=[-infty,infty].

Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.

A function

f:X\to\overline{\R}

is continuous if and only if it is both upper and lower semicontinuous.

The characteristic function or indicator function of a set

A\subsetX

(defined by

1_A(x)=1

x\inA

and

x\notinA

) is upper semicontinuous if and only if

is a closed set. It is lower semicontinuous if and only if

is an open set.

In the field of convex analysis, the characteristic function of a set

A\subsetX

is defined differently, as

\chi_A(x)=0

x\inA

and

\chi_A(x)=infty

x\notinA

. With that definition, the characteristic function of any is lower semicontinuous, and the characteristic function of any is upper semicontinuous.

Binary Operations on Semicontinuous Functions

Let

f,g:X\to\overline{\R}

and

are lower semicontinuous, then the sum

f+g

is lower semicontinuous^[6] (provided the sum is well-defined, i.e.,

f(x)+g(x)

is not the indeterminate form

-infty+infty

). The same holds for upper semicontinuous functions.

and

are lower semicontinuous and non-negative, then the product function

is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.

The function

is lower semicontinuous if and only if

-f

is upper semicontinuous.

and

are upper semicontinuous and

is non-decreasing, then the composition

f\circg

is upper semicontinuous. On the other hand, if

is not non-decreasing, then

f\circg

may not be upper semicontinuous.^[7]

and

are lower semicontinuous, their (pointwise) maximum and minimum (defined by

x\mapstomax\{f(x),g(x)\}

and

x\mapstomin\{f(x),g(x)\}

) are also lower semicontinuous. Consequently, the set of all lower semicontinuous functions from

\overline{\R}

(or to

) forms a lattice. The corresponding statements also hold for upper semicontinuous functions.

Optimization of Semicontinuous Functions

The (pointwise) supremum of an arbitrary family

(f_i)_i\in

of lower semicontinuous functions

f_{i:X\to\overline{\R}}

(defined by

f(x)=\sup\{f_i(x):i\inI\}

) is lower semicontinuous.^[8]

In particular, the limit of a monotone increasing sequence

f_1\lef_2\lef_{3\le …}

of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions

	n
f
	n(x)=1-(1-x)

defined for

x\in[0,1]

for

n=1,2,\ldots.

Likewise, the infimum of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing sequence of continuous functions is upper semicontinuous.

is a compact space (for instance a closed bounded interval

[a,b]

) and

f:C\to\overline{\R}

is upper semicontinuous, then

attains a maximum on

is lower semicontinuous on

it attains a minimum on

(Proof for the upper semicontinuous case: By condition (5) in the definition,

is continuous when

\overline{\R}

is given the left order topology. So its image

f(C)

is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the extreme value theorem.)

Other Properties

(Theorem of Baire)^[9] Let

be a metric space. Every lower semicontinuous function

f:X\to\overline{\R}

is the limit of a point-wise increasing sequence of extended real-valued continuous functions on

In particular, there exists a sequence

\{f_i\}

of continuous functions

f_i:X\to\overline\R

such that

$f_i(x) \leq f_(x) \quad \forall x \in X,\ \forall i = 0, 1, 2, \dots$ and

$\lim_ f_i(x) = f(x) \quad \forall x \in X.$

does not take the value

-infty

, the continuous functions can be taken to be real-valued.^[10] ^[11]

Additionally, every upper semicontinuous function

f:X\to\overline{\R}

is the limit of a monotone decreasing sequence of extended real-valued continuous functions on

does not take the value

infty,

the continuous functions can be taken to be real-valued.

Any upper semicontinuous function

f:X\to\N

on an arbitrary topological space

is locally constant on some dense open subset of

If the topological space

is sequential, then

f:X\toR

is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any

x\inX

and any sequence

(x_n)_n\subsetX

that converges towards

, there holds

\limsup_nf(x_n)\leqslantf(x)

. Equivalently, in a sequential space,

is upper semicontinuous if and only if its superlevel sets

\{x\inX|f(x)\geqslanty\}

are sequentially closed for all

y\inR

. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.

Semicontinuity of Set-valued Functions

For set-valued functions, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower hemicontinuity.A set-valued function

from a set

to a set

is written

F:A\rightrightarrowsB.

For each

x\inA,

the function

defines a set

F(x)\subsetB.

The preimage of a set

S\subsetB

under

is defined as

F^(S) :=\.

That is,

F^-1(S)

is the set that contains every point

such that

F(x)

is not disjoint from

Upper and Lower Semicontinuity

A set-valued map

F:R^m\rightrightarrowsRⁿ

is upper semicontinuous at

x\inR^m

if for every open set

U\subsetRⁿ

such that

F(x)\subsetU

, there exists a neighborhood

such that

F(V)\subsetU.

A set-valued map

F:R^m\rightrightarrowsRⁿ

is lower semicontinuous at

x\inR^m

if for every open set

U\subsetRⁿ

such that

x\inF^-1(U),

there exists a neighborhood

such that

V\subsetF^-1(U).

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing

R^m

and

Rⁿ

in the above definitions with arbitrary topological spaces.

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map. For example, the function

f:R\toR

defined by

f(x) = \begin-1 & \mbox x < 0,\\ 1 & \mbox x \geq 0\end

is upper semicontinuous in the single-valued sense but the set-valued map

x\mapstoF(x):=\{f(x)\}

is not upper semicontinuous in the set-valued sense.

Inner and Outer Semicontinuity

A set-valued function

F:R^m\rightrightarrowsRⁿ

is called inner semicontinuous at

if for every

y\inF(x)

and every convergent sequence

(x_i)

R^m

such that

x_i\tox

, there exists a sequence

(y_i)

Rⁿ

such that

y_i\toy

and

y_i\inF\left(x_i\right)

for all sufficiently large

i\inN.

^[12]

A set-valued function

F:R^m\rightrightarrowsRⁿ

is called outer semicontinuous at

if for every convergence sequence

(x_i)

R^m

such that

x_i\tox

and every convergent sequence

(y_i)

Rⁿ

such that

y_i\inF(x_i)

for each

i\inN,

the sequence

(y_i)

converges to a point in

F(x)

(that is,

\lim_iy_i\inF(x)

Bibliography

Benesova. B.. Kruzik. M.. 2017. Weak Lower Semicontinuity of Integral Functionals and Applications. 10.1137/16M1060947. SIAM Review. 59. 4. 703–766. 1601.00390 . 119668631 .
Book: Bourbaki , Nicolas . Elements of Mathematics: General Topology, 1–4. Springer. 1998. 0-201-00636-7.
Book: Bourbaki , Nicolas . Elements of Mathematics: General Topology, 5–10. Springer. 1998. 3-540-64563-2.
Book: Engelking, Ryszard. Ryszard Engelking. General Topology. Heldermann Verlag, Berlin. 1989. 3-88538-006-4.
Book: Gelbaum , Bernard R. . Olmsted, John M.H.. Counterexamples in analysis. Dover Publications. 2003. 0-486-42875-3.
Book: Hyers , Donald H. . Isac, George . Rassias, Themistocles M.. Topics in nonlinear analysis & applications. World Scientific. 1997. 981-02-2534-2.
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Notes and References

Web site: Verry . Matthieu . Histoire des mathématiques - René Baire .
Stromberg, p. 132, Exercise 4
Book: ((Kurdila, A. J.)), ((Zabarankin, M.)) . 2005 . Convex Functional Analysis . Lower Semicontinuous Functionals . Birkhäuser-Verlag . Systems & Control: Foundations & Applications . 1st . 205–219 . 10.1007/3-7643-7357-1_7 . 978-3-7643-2198-7.
Willard, p. 49, problem 7K
Book: Giaquinta, Mariano . Mathematical analysis : linear and metric structures and continuity . 2007 . Birkhäuser . Giuseppe Modica . 978-0-8176-4514-4 . 1 . Boston . Theorem 11.3, p.396 . 213079540.
Book: Puterman. Martin L.. Markov Decision Processes Discrete Stochastic Dynamic Programming. limited. 2005. Wiley-Interscience. 978-0-471-72782-8. 602.
Book: Moore. James C.. Mathematical methods for economic theory. limited. 1999. Springer. Berlin. 9783540662358. 143.
Web site: To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous .
The result was proved by René Baire in 1904 for real-valued function defined on
\R

. It was extended to metric spaces by Hans Hahn in 1917, and Hing Tong showed in 1952 that the most general class of spaces where the theorem holds is the class of perfectly normal spaces. (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)
Stromberg, p. 132, Exercise 4(g)
Web site: Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions .
In particular, there exists
i₀\geq0

such that
y_i\inF(x_i)

for every natural number
i\geqi_0,

. The necessisty of only considering the tail of
y_i

comes from the fact that for small values of
i,

the set
F(x_i)

may be empty.