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 $$\Gamma : A \to B$$ is said to be upper hemicontinuous at the point $$a$$ if, for any 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 $$V.$$

Sequential characterization
For a set-valued function $$\Gamma : A \to B$$ with closed values, if $$\Gamma : A \to B$$ is upper hemicontinuous at $$a \in A$$ then for all sequences $$a_{\bull} = \left(a_m\right)_{m=1}^{\infty}$$ in $$A$$ and all sequences $$\left(b_m\right)_{m=1}^{\infty}$$ such that $$b_m \in \Gamma\left(a_m\right),$$
 * if $$\lim_{m \to \infty} a_m = a$$ and $$\lim_{m \to \infty} b_m = b$$ then $$b \in \Gamma(a).$$

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).

If B is compact, the converse is also true.

Closed graph theorem
The graph of a set-valued function $$\Gamma : A \to B$$ is the set defined by $$Gr(\Gamma) = \{ (a,b) \in A \times B : b \in \Gamma(a) \}.$$

If $$\Gamma : A \to B$$ is an upper hemicontinuous set-valued function with closed domain (that is, the set of points $$a \in A$$ where $$\Gamma(a)$$ is not the empty set is closed) and closed values (i.e. $$\Gamma(a)$$ is closed for all $$a \in A$$), then $$\operatorname{Gr}(\Gamma)$$ is closed. If $$B$$ is compact, then the converse is also true.

Formal definition: lower hemicontinuity
A set-valued function $$\Gamma : A \to B$$ is said to be lower hemicontinuous at the point $$a$$ if for any 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 $$V$$   $$S$$ means nonempty intersection $$V \cap S \neq \varnothing$$).

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

Open graph theorem
A set-valued function $$\Gamma : A \to B$$ have 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.

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

The open graph theorem says that 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.

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:


 * $$\Gamma : A \to B$$ 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).