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

f

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(x0\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

to

f\left(x0\right)+c

for some

c>0

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

f\left(x0\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

X

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(x0\right)

there exists a neighborhood

U

of

x0

such that

f(x)<y

for all

x\inU

.[2] Equivalently,

f

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

f

at the point

x0

.

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

X

, where

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

.

(3) For each

y\in\R

, the

y

-superlevel set

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

is closed in

X

.

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

is closed in

X x \R

.

(5) The function

f

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

x0\inX

if for every real

y<f\left(x0\right)

there exists a neighborhood

U

of

x0

such that

f(x)>y

for all

x\inU

.Equivalently,

f

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

f

at point

x0

.

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

X

, where

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

.

(3) For each

y\in\R

, the

y

-sublevel set

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

is closed in

X

.

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

is closed in

X x \R

.[3]

(5) The function

f

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]

.

Semicontinuity in Metric Spaces

When

X

is a metric space, then upper and lower semicontinuity can be stated using a

\delta

-

\varepsilon

formulation, similar to continuity.In particular, if

X

is a metric space, then

f:X\to\R\cup\{-infty\}

is upper semi-continuous at

x0\inX

if for all

\varepsilon>0,

there exists

\delta>0

such that

B\left(x0,\delta\right)

is a ball of radius

\delta

(relative to the metric) centered at

x0.

Similarly,

f:X\to\R\cup\{+infty\}

is lower semi-continuous at

x0\inX

if for all

\varepsilon>0,

there exists

\delta>0

such that

B\left(x0,\delta\right)

is a ball of radius

\delta

(relative to the metric) centered at

x0.

[4]

Examples

Consider the function

f,

piecewise defined by:f(x) = \begin-1 & \mbox x < 0,\\ 1 & \mbox x \geq 0\endThis 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

x,

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.[5] For example the functionf(x) = \begin\sin(1/x) & \mbox x \neq 0,\\1 & \mbox x = 0,\endis upper semicontinuous at

x=0

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

If

X=\Rn

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

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

is the space of curves in

X

(with the supremum distance

d\Gamma(\alpha,\beta)=\sup\{dX(\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.[6] 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)

to

[-infty,+infty]

is lower semicontinuous.

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

Properties

X

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.

f:X\to\overline{\R}

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

A\subsetX

(defined by

1A(x)=1

if

x\inA

and

0

if

x\notinA

) is upper semicontinuous if and only if

A

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

A

is an open set.

A\subsetX

is defined differently, as

\chiA(x)=0

if

x\inA

and

\chiA(x)=infty

if

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}

.

f

and

g

are lower semicontinuous, then the sum

f+g

is lower semicontinuous[7] (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.

f

and

g

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

fg

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

f

is lower semicontinuous if and only if

-f

is upper semicontinuous.

f

and

g

are upper semicontinuous and

f

is non-decreasing, then the composition

f\circg

is upper semicontinuous. On the other hand, if

f

is not non-decreasing, then

f\circg

may not be upper semicontinuous.[8]

f

and

g

are lower semicontinuous, then the functions

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

and

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

are also lower semicontinuous. Similarly, if

f

and

g

are upper semicontinuous, then

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

and

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

are upper semicontinuous.

X

to

\overline{\R}

(or to

\R

) forms a lattice. This follows directly from the fact that each pair of lower semicontinuous functions

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

has a maximum and minimum that is also lower semicontinuous. Similarly, the set of all upper semicontinuous functions also form a lattice.

Optimization of Semicontinuous Functions

(fi)i\in

of lower semicontinuous functions

fi:X\to\overline{\R}

(defined by

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

) is lower semicontinuous.[9]

In particular, the limit of a monotone increasing sequence

f1\lef2\lef3\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.

C

is a compact space (for instance a closed bounded interval

[a,b]

) and

f:C\to\overline{\R}

is upper semicontinuous, then

f

attains a maximum on

C.

If

f

is lower semicontinuous on

C,

it attains a minimum on

C.

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

f

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

X

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

X.

In particular, there exists a sequence

\{fi\}

of continuous functions

fi: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.

If

f

does not take the value

-infty

, the continuous functions can be taken to be real-valued.[11] [12]

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

X;

if

f

does not take the value

infty,

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

f:X\to\N

on an arbitrary topological space

X

is locally constant on some dense open subset of

X.

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

F

from a set

A

to a set

B

is written

F:A\rightrightarrowsB.

For each

x\inA,

the function

F

defines a set

F(x)\subsetB.

The preimage of a set

S\subsetB

under

F

is defined as F^(S) :=\.That is,

F-1(S)

is the set that contains every point

x

in

A

such that

F(x)

is not disjoint from

S

.

Upper and Lower Semicontinuity

A set-valued map

F:Rm\rightrightarrowsRn

is upper semicontinuous at

x\inRm

if for every open set

U\subsetRn

such that

F(x)\subsetU

, there exists a neighborhood

V

of

x

such that

F(V)\subsetU.

A set-valued map

F:Rm\rightrightarrowsRn

is lower semicontinuous at

x\inRm

if for every open set

U\subsetRn

such that

x\inF-1(U),

there exists a neighborhood

V

of

x

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

Rm

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\endis 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:Rm\rightrightarrowsRn

is called inner semicontinuous at

x

if for every

y\inF(x)

and every convergent sequence

(xi)

in

Rm

such that

xi\tox

, there exists a sequence

(yi)

in

Rn

such that

yi\toy

and

yi\inF\left(xi\right)

for all sufficiently large

i\inN.

[13]

A set-valued function

F:Rm\rightrightarrowsRn

is called outer semicontinuous at

x

if for every convergence sequence

(xi)

in

Rm

such that

xi\tox

and every convergent sequence

(yi)

in

Rn

such that

yi\inF(xi)

for each

i\inN,

the sequence

(yi)

converges to a point in

F(x)

(that is,

\limiyi\inF(x)

).

Bibliography

Notes and References

  1. Web site: Verry . Matthieu . Histoire des mathématiques - René Baire .
  2. Stromberg, p. 132, Exercise 4
  3. 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.
  4. Book: ((Aubin, J. P.)) . 1993 . Optima and equilibria : an introduction to nonlinear analysis . Berlin ; New York : Springer-Verlag . 978-3-540-52121-1.
  5. Willard, p. 49, problem 7K
  6. 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.
  7. Book: Puterman. Martin L.. Markov Decision Processes Discrete Stochastic Dynamic Programming. limited. 2005. Wiley-Interscience. 978-0-471-72782-8. 602.
  8. Book: Moore. James C.. Mathematical methods for economic theory. limited. 1999. Springer. Berlin. 9783540662358. 143.
  9. Web site: To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous .
  10. 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.)
  11. Stromberg, p. 132, Exercise 4(g)
  12. Web site: Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions .
  13. In particular, there exists

    i0\geq0

    such that

    yi\inF(xi)

    for every natural number

    i\geqi0,

    . The necessisty of only considering the tail of

    yi

    comes from the fact that for small values of

    i,

    the set

    F(xi)

    may be empty.

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