Upper hemicontinuity is approximately when the graph of the correspondence is closed from the left and from the right (but not necessarily closed at every point). And Lower hemicontinuity is approximately when the graph has no closed edges (i.e. a point that a convergent sequence in the range can reach from the left or right, but not from the other direction).
A correspondence Γ : A → B is said to be upper hemicontinuous at the point a if for any open neighbourhood V of Γ(a) there exists neighbourhood U of a s.t. Γ(x) is in V for all x in U.
One should be warned that the term upper semicontinuous, instead of upper hemicontinuous, has been more widespreaded in the literature. Then upper hemicontinuous is reserved for upper semicontinuity w.r.t. weak topology.
Multivalued Closed Graph Theorem: For correspondence Γ : A → B with closed values (i.e. Γ (a) - closed for a in A) and compact range (i.e. is compact subset of B) to be upper hemicontinuous it is sufficient and necessary to have closed graph in A×B; sequentially:
A correspondence Γ : A → B is said to be lower hemicontinuous at the point a if for any open set V intersecting Γ(a) there exists neighbourhood U of a s.t. Γ(x) intersects V for all x in U. (Here V intersects S means nonempty intersection ).
One should be warned that the term lower semicontinuous, instead of lower hemicontinuous, has been more widespreaded in the literature. Then lower hemicontinuous is reserved for lower semicontinuity w.r.t. weak topology.
Sequential characterization: Γ : A → B is lower hemicontinuous at a if and only if
Open Graph Theorem: If Γ : A → B has open graph Gr(Γ), then it is lower hemicontinuous.
Very important part of set-valued analysis (in view of applications) constitutes the investigation of single-valued selections and approximations to multivalued maps. Typically lower hemicontinuous correspondences 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).
Γ : A → B is lower [resp. upper] hemicontinuous if and only if the mapping
Γ : A → P(B) is continuous where the hyperspace P(B) has been endowed with the lower
[resp. upper] Vietoris topology.
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).