Английская Википедия:Hemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A set-valued function that has both properties is said to be continuous in an analogy to the property of the same name for single-valued functions.
To explain both notions, consider a sequence a of points in a domain, and a sequence b of points in the range. We say that b corresponds to a if each point in b is contained in the image of the corresponding point in a.
- Upper hemicontinuity requires that, for any convergent sequence a in a domain, and for any convergent sequence b that corresponds to a, the image of the limit of a contains the limit of b.
- Lower hemicontinuity requires that, for any convergent sequence a in a domain, and for any point x in the image of the limit of a, there exists a sequence b that corresponds to a subsequence of a, that converges to x.
Examples
The image on the right shows a function that is not lower hemicontinuous at x. To see this, let a be a sequence that converges to x from the left. The image of x is a vertical line that contains some point (x,y). But every sequence b that corresponds to a is contained in the bottom horizontal line, so it cannot converge to y. In contrast, the function is upper hemicontinuous everywhere. For example, considering any sequence a that converges to x from the left or from the right, and any corresponding sequence b, the limit of b is contained in the vertical line that is the image of the limit of a.
The image on the left shows a function that is not upper hemicontinuous at x. To see this, let a be a sequence that converges to x from the right. The image of a contains vertical lines, so there exists a corresponding sequence b in which all elements are bounded away from f(x). The image of the limit of a contains a single point f(x), so it does not contain the limit of b. In contrast, that function is lower hemicontinuous everywhere. For example, for any sequence a that converges to x, from the left or from the right, f(x) contains a single point, and there exists a corresponding sequence b that converges to f(x).
Formal definition: upper hemicontinuity
A set-valued function <math>\Gamma : A \to B</math> is said to be upper hemicontinuous at the point <math>a</math> if, for any open <math>V \subset B </math> with <math> \Gamma(a) \subset V</math>, there exists a neighbourhood <math>U</math> of <math>a</math> such that for all <math>x \in U,</math> <math>\Gamma(x)</math> is a subset of <math>V.</math>
Sequential characterization
For a set-valued function <math>\Gamma : A \to B</math> with closed values, if <math>\Gamma : A \to B</math> is upper hemicontinuous at <math>a \in A</math> then for all sequences <math>a_{\bull} = \left(a_m\right)_{m=1}^{\infty}</math> in <math>A</math> and all sequences <math>\left(b_m\right)_{m=1}^{\infty}</math> such that <math>b_m \in \Gamma\left(a_m\right),</math>
- if <math>\lim_{m \to \infty} a_m = a</math> and <math>\lim_{m \to \infty} b_m = b</math> then <math>b \in \Gamma(a).</math>
As an example, look at the image at the right, and consider sequence a in the domain that converges to x (either from the left or from the right). Then, any sequence b that satisfies the requirements converges to some point in f(x). Therefore, the
If B is compact, the converse is also true.
Closed graph theorem
The graph of a set-valued function <math>\Gamma : A \to B</math> is the set defined by <math>Gr(\Gamma) = \{ (a,b) \in A \times B : b \in \Gamma(a) \}.</math>
If <math>\Gamma : A \to B</math> is an upper hemicontinuous set-valued function with closed domain (that is, the set of points <math>a \in A</math> where <math>\Gamma(a)</math> is not the empty set is closed) and closed values (i.e. <math>\Gamma(a)</math> is closed for all <math>a \in A</math>), then <math>\operatorname{Gr}(\Gamma)</math> is closed. If <math>B</math> is compact, then the converse is also true.[1]
Formal definition: lower hemicontinuity
A set-valued function <math>\Gamma : A \to B</math> is said to be lower hemicontinuous at the point <math>a</math> if for any open set <math>V</math> intersecting <math>\Gamma(a)</math> there exists a neighbourhood <math>U</math> of <math>a</math> such that <math>\Gamma(x)</math> intersects <math>V</math> for all <math>x \in U.</math> (Here <math>V</math> Шаблон:Em <math>S</math> means nonempty intersection <math>V \cap S \neq \varnothing</math>).
Sequential characterization
<math>\Gamma : A \to B</math> is lower hemicontinuous at <math>a</math> if and only if for every sequence <math>a_{\bull} = \left(a_m\right)_{m=1}^{\infty}</math> in <math>A</math> such that <math>a_{\bull} \to a</math> in <math>A</math> and all <math>b \in \Gamma(a),</math> there exists a subsequence <math>\left(a_{m_k}\right)_{k=1}^{\infty}</math> of <math>a_{\bull}</math> and also a sequence <math>b_{\bull} = \left(b_k\right)_{k=1}^{\infty}</math> such that <math>b_{\bull} \to b</math> and <math>b_k \in \Gamma\left(a_{m_k}\right)</math> for every <math>k.</math>
Open graph theorem
A set-valued function <math>\Gamma : A \to B</math> have Шаблон:Em if the set <math>\Gamma^{-1}(b) = \{ a \in A : b \in \Gamma(a) \}</math> is open in <math>A</math> for every <math>b \in B.</math> If <math>\Gamma</math> values are all open sets in <math>B,</math> then <math>\Gamma</math> is said to have Шаблон:Em.
If <math>\Gamma</math> has an open graph <math>\operatorname{Gr}(\Gamma),</math> then <math>\Gamma</math> has open upper and lower sections and if <math>\Gamma</math> has open lower sections then it is lower hemicontinuous.[2]
The open graph theorem says that if <math>\Gamma : A \to P\left(\R^n\right)</math> is a set-valued function with convex values and open upper sections, then <math>\Gamma</math> has an open graph in <math>A \times \R^n</math> if and only if <math>\Gamma</math> is lower hemicontinuous.[2]
Properties
Set-theoretic, algebraic and topological operations on set-valued functions (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 set-valued functions 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 set-valued functions. Typically lower hemicontinuous set-valued functions 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 set-valued function 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:
- <math>\Gamma : A \to B</math> is lower [resp. upper] hemicontinuous if and only if the mapping <math>\Gamma : A \to P(B)</math> 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
References
- Шаблон:Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
Шаблон:Convex analysis and variational analysis
- ↑ Proposition 1.4.8 of Шаблон:Cite book
- ↑ 2,0 2,1 Шаблон:Cite journal
