graph theorem says that if is a set-valued function with convex values and open upper sections, then has an open graph in if and only if 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: is lower [resp. upper] hemicontinuous if and only if the mapping 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 Theory of continuous functions Mathematical analysis Variational analysis