Hemicontinuity

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

This set-valued function is upper hemicontinuous everywhere, but not lower hemicontinuous at $x$ : for a sequence of points $\left(x_m\right)$ that converges to $x,$ we have a $y$ ($y\in f(x)$) such that no sequence of $\left(y_m\right)$ converges to $y$ where each $y_m$ is in $f\left(x_m\right).$

This set-valued function is lower hemicontinuous everywhere, but not upper hemicontinuous at $x,$ because the graph (set) is not closed.

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 $\Gamma :A\rightrightarrows B$ is said to be upper hemicontinuous at a point $a\in A$ if, for every open $V\subset B$ with $\Gamma (a)\subset V,$ there exists a neighbourhood $U$ of $a$ such that for all $x\in U,$ $\Gamma (x)$ is a subset of $V.$

Lower hemicontinuity

A set-valued function $\Gamma :A\rightrightarrows B$ is said to be lower hemicontinuous at the point $a\in A$ if for every open set $V$ intersecting $\Gamma (a),$ there exists a neighbourhood $U$ of $a$ such that $\Gamma (x)$ intersects $V$ for all $x\in U.$ (Here $V$ intersects $S$ means nonempty intersection $V\cap S\ne \varnothing$).

Continuity

If a set-valued function is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous.

Properties

Upper hemicontinuity

Sequential characterization

Theorem—For a set-valued function $\Gamma :A\rightrightarrows B$ with closed values, if $\Gamma$ is upper hemicontinuous at $a\in A,$ then for every sequence $a_{\bullet }={\left(a_m\right)}_{m=1}^{\mathrm{\infty }}$ in $A$ and every sequence ${\left(b_m\right)}_{m=1}^{\mathrm{\infty }}$ such that $b_m\in \Gamma \left(a_m\right),$

if $\underset{m\to \mathrm{\infty }}{\lim }{a}_m=a$ and $\underset{m\to \mathrm{\infty }}{\lim }{b}_m=b$ then $b\in \Gamma (a).$

If $B$ is compact, then the converse is also true.

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 $\Gamma :A\rightrightarrows B$ is the set defined by $Gr(\Gamma )=\{(a,b)\in A\times B:b\in \Gamma (a)\}.$ The domain of $\Gamma$ is the set of all $a\in A$ such that $\Gamma (a)$ is not empty.

Theorem—If $\Gamma :A\rightrightarrows B$ is an upper hemicontinuous set-valued function with closed domain (that is, the domain of $\Gamma$ is closed) and closed values (i.e. $\Gamma (a)$ is closed for all $a\in A$), then $\mathrm{Gr}(\Gamma )$ is closed.

If $B$ is compact, then the converse is also true.

Lower hemicontinuity

Sequential characterization

Theorem—$\Gamma :A\rightrightarrows B$ is lower hemicontinuous at $a\in A$ if and only if for every sequence $a_{\bullet }={\left(a_m\right)}_{m=1}^{\mathrm{\infty }}$ in $A$ such that $a_{\bullet }\to a$ in $A$ and all $b\in \Gamma (a),$ there exists a subsequence ${\left(a_{m_k}\right)}_{k=1}^{\mathrm{\infty }}$ of $a_{\bullet }$ and also a sequence $b_{\bullet }={\left(b_k\right)}_{k=1}^{\mathrm{\infty }}$ such that $b_{\bullet }\to b$ and $b_k\in \Gamma \left(a_{m_k}\right)$ for every $k.$

Open graph theorem

A set-valued function $\Gamma :A\to B$ is said to have open lower sections if the set ${\Gamma }^{-1}(b)=\{a\in A:b\in \Gamma (a)\}$ is open in $A$ for every $b\in B.$ If $\Gamma$ values are all open sets in $B,$ then $\Gamma$ is said to have open upper sections.

If $\Gamma$ has an open graph $\mathrm{Gr}(\Gamma ),$ then $\Gamma$ has open upper and lower sections and if $\Gamma$ has open lower sections then it is lower hemicontinuous.

Open Graph Theorem—If $\Gamma :A\to P\left({ℝ}^n\right)$ is a set-valued function with convex values and open upper sections, then $\Gamma$ has an open graph in $A\times {ℝ}^n$ if and only if $\Gamma$ is lower hemicontinuous.

Operations Preserving Hemicontinuity

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.

Function Selections

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).

Other concepts of continuity

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

Theorem— A set-valued map $\Gamma :A\to B$ is lower [resp. upper] hemicontinuous if and only if the mapping $\Gamma :A\to P(B)$ 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

• Hausdorff distance – Distance between two metric-space subsets
• Semicontinuity – Property of functions which is weaker than continuityPages displaying short descriptions of redirect targets
• Selection theorem - a theorem about constructing a single-valued function from a set-valued function.