Hemicontinuity

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 S means nonempty intersection 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 {{Math 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_{\bull} = \left(a_m\right)_{m=1}^{\infty} in A and every sequence \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). 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. {{Math 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 \operatorname{Gr}(\Gamma) is closed. If B is compact, then the converse is also true. }} Lower hemicontinuity Sequential characterization {{Math theorem|\Gamma : A \rightrightarrows B is lower hemicontinuous at a \in 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 is said to 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. 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: (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==