Hemicontinuity

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In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions.

Contents

Roughly speaking, a function is upper hemicontinuous when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.

Upper hemicontinuity

This correspondence is upper hemicontinuous everywhere, but not lower hemicontinuous at
x
{\displaystyle x}
: for a sequence of points
(
x
m
)
{\displaystyle \left(x_{m}\right)}
that converges to
x
,
{\displaystyle x,}
we have a
y
{\displaystyle y}
(
y
[?]
f
(
x
)
{\displaystyle y\in f(x)}
) such that no sequence of
(
y
m
)
{\displaystyle \left(y_{m}\right)}
converges to
y
{\displaystyle y}
where each
y
m
{\displaystyle y_{m}}
is in
f
(
x
m
)
.
{\displaystyle f\left(x_{m}\right).} Upper hemicontinuous.svg
This correspondence is upper hemicontinuous everywhere, but not lower hemicontinuous at : for a sequence of points that converges to we have a () such that no sequence of converges to where each is in

A correspondence is said to be upper hemicontinuous at the point if for any open neighbourhood of there exists a neighbourhood of such that for all is a subset of

Sequential characterization

For a correspondence with closed values, if is upper hemicontinuous at then for all sequences in for all all sequences such that

if and then

If B is compact, the converse is also true.

Closed graph theorem

The graph of a correspondence is the set defined by

If is an upper hemicontinuous correspondence with closed domain (that is, the set of points where is not the empty set is closed) and closed values (i.e. is closed for all ), then is closed. If is compact, then the converse is also true. [1]

Lower hemicontinuity

This correspondence is lower hemicontinuous everywhere, but not upper hemicontinuous at
x
,
{\displaystyle x,}
because the graph (set) is not closed. Lower hemicontinuous.svg
This correspondence is lower hemicontinuous everywhere, but not upper hemicontinuous at because the graph (set) is not closed.

A correspondence is said to be lower hemicontinuous at the point if for any open set intersecting there exists a neighbourhood of such that intersects for all (Here intersects means nonempty intersection ).

Sequential characterization

is lower hemicontinuous at if and only if for every sequence in such that in and all there exists a subsequence of and also a sequence such that and for every

Open graph theorem

A correspondence have open lower sections if the set is open in for every If values are all open sets in then is said to have open upper sections.

If has an open graph then has open upper and lower sections and if has open lower sections then it is lower hemicontinuous. [2]

The open graph theorem says that if is a convex-valued correspondence with open upper sections, then has an open graph in if and only if is lower hemicontinuous. [2]

Properties

Set-theoretic, algebraic and topological operations on multivalued maps (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous correspondences whose intersection is not lower hemicontinuous. This can be fixed upon strengthening continuity properties: if one of those lower hemicontinuous multifunctions has open graph then their intersection is again lower hemicontinuous.

Crucial to set-valued analysis (in view of applications) are the investigation of single-valued selections and approximations to multivalued maps. Typically lower hemicontinuous correspondences admit single-valued selections (Michael selection theorem, Bressan–Colombo directionally continuous selection theorem, Fryszkowski decomposable map selection). Likewise, upper hemicontinuous maps admit approximations (e.g. Ancel–Granas–Górniewicz–Kryszewski theorem).

Implications for continuity

If a correspondence is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous. A continuous function is in all cases both upper and lower hemicontinuous.

Other concepts of continuity

The upper and lower hemicontinuity might be viewed as usual continuity:

is lower [resp. upper] hemicontinuous if and only if the mapping is continuous where the hyperspace P(B) has been endowed with the lower [resp. upper] Vietoris topology.

(For the notion of hyperspace compare also power set and function space).

Using lower and upper Hausdorff uniformity we can also define the so-called upper and lower semicontinuous maps in the sense of Hausdorff (also known as metrically lower / upper semicontinuous maps).

See also

Notes

  1. Proposition 1.4.8 of Aubin, Jean-Pierre; Frankowska, Hélène (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN   3-7643-3478-9.
  2. 1 2 Zhou, J.X. (August 1995). "On the Existence of Equilibrium for Abstract Economies". Journal of Mathematical Analysis and Applications. 193 (3): 839–858. doi:10.1006/jmaa.1995.1271.

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