In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets *A* and *B*. The dual concepts of **upper hemicontinuity** and **lower hemicontinuity** facilitate such an extension. A correspondence that has both properties is said to be **continuous** in an analogy to the property of the same name for functions.

- Upper hemicontinuity
- Sequential characterization
- Closed graph theorem
- Lower hemicontinuity
- Sequential characterization 2
- Open graph theorem
- Properties
- Implications for continuity
- Other concepts of continuity
- See also
- Notes
- References

Roughly speaking, a function is upper hemicontinuous when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.

A correspondence is said to be **upper hemicontinuous** at the point if for any open neighbourhood of there exists a neighbourhood of such that for all is a subset of

For a correspondence with closed values, if is upper hemicontinuous at then for all sequences in for all all sequences such that

- if and then

If B is compact, the converse is also true.

The graph of a correspondence is the set defined by

If is an upper hemicontinuous correspondence with closed domain (that is, the set of points where is not the empty set is closed) and closed values (i.e. is closed for all ), then is closed. If is compact, then the converse is also true.^{ [1] }

A correspondence is said to be **lower hemicontinuous** at the point if for any open set intersecting there exists a neighbourhood of such that intersects for all (Here *intersects* means nonempty intersection ).

is lower hemicontinuous at if and only if for every sequence in such that in and all there exists a subsequence of and also a sequence such that and for every

A correspondence have *open lower sections* if the set is open in for every If values are all open sets in then is said to have *open upper sections*.

If has an open graph then has open upper and lower sections and if has open lower sections then it is lower hemicontinuous.^{ [2] }

The open graph theorem says that if is a convex-valued correspondence with open upper sections, then has an open graph in if and only if is lower hemicontinuous.^{ [2] }

Set-theoretic, algebraic and topological operations on multivalued maps (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 correspondences 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 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).

If a correspondence 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.

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**).

- Differential inclusion
- Hausdorff distance
- Multivalued function – Generalization of a function that may produce several outputs for each input
- Semicontinuity

- ↑ Proposition 1.4.8 of Aubin, Jean-Pierre; Frankowska, Hélène (1990).
*Set-Valued Analysis*. Basel: Birkhäuser. ISBN 3-7643-3478-9. - 1 2 Zhou, J.X. (August 1995). "On the Existence of Equilibrium for Abstract Economies".
*Journal of Mathematical Analysis and Applications*.**193**(3): 839–858. doi:10.1006/jmaa.1995.1271.

The **Cauchy distribution**, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the **Lorentz distribution**, **Cauchy–Lorentz distribution**, **Lorentz(ian) function**, or **Breit–Wigner distribution**. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

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In functional analysis, a branch of mathematics, **Michael selection theorem** is a selection theorem named after Ernest Michael. In its most popular form, it states the following:

The **maximum theorem** provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control.

- Aliprantis, Charalambos D.; Border, Kim C. (2007).
*Infinite Dimensional Analysis: Hitchhiker's Guide*(Third ed.). Berlin: Springer. ISBN 978-3-540-32696-0. - Aubin, Jean-Pierre; Cellina, Arrigo (1984).
*Differential Inclusions: Set-Valued Maps and Viability Theory*. Grundl. der Math. Wiss.**264**. Berlin: Springer. ISBN 0-387-13105-1. - Aubin, Jean-Pierre; Frankowska, Hélène (1990).
*Set-Valued Analysis*. Basel: Birkhäuser. ISBN 3-7643-3478-9. - Deimling, Klaus (1992).
*Multivalued Differential Equations*. Walter de Gruyter. ISBN 3-11-013212-5. - Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995).
*Microeconomic Analysis*. New York: Oxford University Press. pp. 949–951. ISBN 0-19-507340-1. - Ok, Efe A. (2007).
*Real Analysis with Economic Applications*. Princeton University Press. pp. 216–226. ISBN 0-691-11768-3.

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