In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of upper and lower semicontinuity of single-valued functions to set-valued functions. A set-valued function that is both upper and lower hemicontinuous 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).
Definitions
Upper hemicontinuity A set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is said to be upper hemicontinuous at a point a ∈ A {\displaystyle a\in A} if, for every open V ⊂ B {\displaystyle V\subset B} with Γ ( a ) ⊂ V , {\displaystyle \Gamma (a)\subset V,} there exists a neighbourhood U {\displaystyle U} of a {\displaystyle a} such that for all x ∈ U , {\displaystyle x\in U,} Γ ( x ) {\displaystyle \Gamma (x)} is a subset of V . {\displaystyle V.}
Lower hemicontinuity A set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is said to be lower hemicontinuous at the point a ∈ A {\displaystyle a\in A} if for every open set V {\displaystyle V} intersecting Γ ( a ) , {\displaystyle \Gamma (a),} there exists a neighbourhood U {\displaystyle U} of a {\displaystyle a} such that Γ ( x ) {\displaystyle \Gamma (x)} intersects V {\displaystyle V} for all x ∈ U . {\displaystyle x\in U.} (Here V {\displaystyle V} intersects S {\displaystyle S} means nonempty intersection V ∩ S ≠ ∅ {\displaystyle V\cap S\neq \varnothing } ).
Continuity If a set-valued function is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous.
Properties
Upper hemicontinuity
Sequential characterization
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).
Closed graph theorem The graph of a set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is the set defined by G r ( Γ ) = { ( a , b ) ∈ A × B : b ∈ Γ ( a ) } . {\displaystyle Gr(\Gamma )=\{(a,b)\in A\times B:b\in \Gamma (a)\}.}
The domain of Γ {\displaystyle \Gamma } is the set of all a ∈ A {\displaystyle a\in A} such that Γ ( a ) {\displaystyle \Gamma (a)} is not empty.
Lower hemicontinuity
Sequential characterization
Open graph theorem A set-valued function Γ : A → B {\displaystyle \Gamma :A\to B} is said to have open lower sections if the set Γ − 1 ( b ) = { a ∈ A : b ∈ Γ ( a ) } {\displaystyle \Gamma ^{-1}(b)=\{a\in A:b\in \Gamma (a)\}}
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