In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than

A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point to for some , then the result is upper semicontinuous; if we decrease its value to then the result is lower semicontinuous.

A function that is upper semicontinuous, but not lower semicontinuous, at . The solid blue dot indicates
A function that is lower semicontinuous, but not upper semicontinuous, at . The solid blue dot indicates

The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.[1]

Definitions

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Assume throughout that   is a topological space and   is a function with values in the extended real numbers  .

Upper semicontinuity

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A function   is called upper semicontinuous at a point   if for every real   there exists a neighborhood   of   such that   for all  .[2] Equivalently,   is upper semicontinuous at   if and only if   where lim sup is the limit superior of the function   at the point  

If   is a metric space with distance function   and   this can also be restated using an  -  formulation, similar to the definition of continuous function. Namely, for each   there is a   such that   whenever  

A function   is called upper semicontinuous if it satisfies any of the following equivalent conditions:[2]

(1) The function is upper semicontinuous at every point of its domain.
(2) For each  , the set   is open in  , where  .
(3) For each  , the  -superlevel set   is closed in  .
(4) The hypograph   is closed in  .
(5) The function   is continuous when the codomain   is given the left order topology. This is just a restatement of condition (2) since the left order topology is generated by all the intervals  .

Lower semicontinuity

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A function   is called lower semicontinuous at a point   if for every real   there exists a neighborhood   of   such that   for all  . Equivalently,   is lower semicontinuous at   if and only if   where   is the limit inferior of the function   at point  

If   is a metric space with distance function   and   this can also be restated as follows: For each   there is a   such that   whenever  

A function   is called lower semicontinuous if it satisfies any of the following equivalent conditions:

(1) The function is lower semicontinuous at every point of its domain.
(2) For each  , the set   is open in  , where  .
(3) For each  , the  -sublevel set   is closed in  .
(4) The epigraph   is closed in  .[3]: 207 
(5) The function   is continuous when the codomain   is given the right order topology. This is just a restatement of condition (2) since the right order topology is generated by all the intervals  .

Examples

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Consider the function   piecewise defined by:   This function is upper semicontinuous at   but not lower semicontinuous.

The floor function   which returns the greatest integer less than or equal to a given real number   is everywhere upper semicontinuous. Similarly, the ceiling function   is lower semicontinuous.

Upper and lower semicontinuity bear no relation to continuity from the left or from the right for functions of a real variable. Semicontinuity is defined in terms of an ordering in the range of the functions, not in the domain.[4] For example the function   is upper semicontinuous at   while the function limits from the left or right at zero do not even exist.

If   is a Euclidean space (or more generally, a metric space) and   is the space of curves in   (with the supremum distance  ), then the length functional   which assigns to each curve   its length   is lower semicontinuous.[5] As an example, consider approximating the unit square diagonal by a staircase from below. The staircase always has length 2, while the diagonal line has only length  .

Let   be a measure space and let   denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to   Then by Fatou's lemma the integral, seen as an operator from   to   is lower semicontinuous.

Tonelli's theorem in functional analysis characterizes the weak lower semicontinuity of nonlinear functionals on Lp spaces in terms of the convexity of another function.

Properties

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Unless specified otherwise, all functions below are from a topological space   to the extended real numbers   Several of the results hold for semicontinuity at a specific point, but for brevity they are only stated for semicontinuity over the whole domain.

  • A function   is continuous if and only if it is both upper and lower semicontinuous.
  • The characteristic function or indicator function of a set   (defined by   if   and   if  ) is upper semicontinuous if and only if   is a closed set. It is lower semicontinuous if and only if   is an open set.
  • In the field of convex analysis, the characteristic function of a set   is defined differently, as   if   and   if  . With that definition, the characteristic function of any closed set is lower semicontinuous, and the characteristic function of any open set is upper semicontinuous.

Binary Operations on Semicontinuous Functions

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Let  .

  • If   and   are lower semicontinuous, then the sum   is lower semicontinuous[6] (provided the sum is well-defined, i.e.,   is not the indeterminate form  ). The same holds for upper semicontinuous functions.
  • If   and   are lower semicontinuous and non-negative, then the product function   is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
  • The function   is lower semicontinuous if and only if   is upper semicontinuous.
  • If   and   are upper semicontinuous and   is non-decreasing, then the composition   is upper semicontinuous. On the other hand, if   is not non-decreasing, then   may not be upper semicontinuous.[7]
  • If   and   are lower semicontinuous, their (pointwise) maximum and minimum (defined by   and  ) are also lower semicontinuous. Consequently, the set of all lower semicontinuous functions from   to   (or to  ) forms a lattice. The corresponding statements also hold for upper semicontinuous functions.

Optimization of Semicontinuous Functions

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  • The (pointwise) supremum of an arbitrary family   of lower semicontinuous functions   (defined by  ) is lower semicontinuous.[8]
In particular, the limit of a monotone increasing sequence   of continuous functions is lower semicontinuous. (The Theorem of Baire below provides a partial converse.) The limit function will only be lower semicontinuous in general, not continuous. An example is given by the functions   defined for   for  
Likewise, the infimum of an arbitrary family of upper semicontinuous functions is upper semicontinuous. And the limit of a monotone decreasing sequence of continuous functions is upper semicontinuous.
  • If   is a compact space (for instance a closed bounded interval  ) and   is upper semicontinuous, then   attains a maximum on   If   is lower semicontinuous on   it attains a minimum on  
(Proof for the upper semicontinuous case: By condition (5) in the definition,   is continuous when   is given the left order topology. So its image   is compact in that topology. And the compact sets in that topology are exactly the sets with a maximum. For an alternative proof, see the article on the extreme value theorem.)

Other Properties

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  • (Theorem of Baire)[note 1] Let   be a metric space. Every lower semicontinuous function   is the limit of a point-wise increasing sequence of extended real-valued continuous functions on   In particular, there exists a sequence   of continuous functions   such that
  and
 
If   does not take the value  , the continuous functions can be taken to be real-valued.[9][10]
Additionally, every upper semicontinuous function   is the limit of a monotone decreasing sequence of extended real-valued continuous functions on   if   does not take the value   the continuous functions can be taken to be real-valued.
  • Any upper semicontinuous function   on an arbitrary topological space   is locally constant on some dense open subset of  
  • If the topological space   is sequential, then   is upper semi-continuous if and only if it is sequentially upper semi-continuous, that is, if for any   and any sequence   that converges towards  , there holds  . Equivalently, in a sequential space,   is upper semicontinuous if and only if its superlevel sets   are sequentially closed for all  . In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.

Semicontinuity of Set-valued Functions

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For set-valued functions, several concepts of semicontinuity have been defined, namely upper, lower, outer, and inner semicontinuity, as well as upper and lower hemicontinuity. A set-valued function   from a set   to a set   is written   For each   the function   defines a set   The preimage of a set   under   is defined as   That is,   is the set that contains every point   in   such that   is not disjoint from  .[11]

Upper and Lower Semicontinuity

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A set-valued map   is upper semicontinuous at   if for every open set   such that  , there exists a neighborhood   of   such that  [11]: Def. 2.1 

A set-valued map   is lower semicontinuous at   if for every open set   such that   there exists a neighborhood   of   such that  [11]: Def. 2.2 

Upper and lower set-valued semicontinuity are also defined more generally for a set-valued maps between topological spaces by replacing   and   in the above definitions with arbitrary topological spaces.[11]

Note, that there is not a direct correspondence between single-valued lower and upper semicontinuity and set-valued lower and upper semicontinuouty. An upper semicontinuous single-valued function is not necessarily upper semicontinuous when considered as a set-valued map.[11]: 18  For example, the function   defined by   is upper semicontinuous in the single-valued sense but the set-valued map   is not upper semicontinuous in the set-valued sense.

Inner and Outer Semicontinuity

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A set-valued function   is called inner semicontinuous at   if for every   and every convergent sequence   in   such that  , there exists a sequence   in   such that   and   for all sufficiently large  [12][note 2]

A set-valued function   is called outer semicontinuous at   if for every convergence sequence   in   such that   and every convergent sequence   in   such that   for each   the sequence   converges to a point in   (that is,  ).[12]

See also

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Notes

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  1. ^ The result was proved by René Baire in 1904 for real-valued function defined on  . It was extended to metric spaces by Hans Hahn in 1917, and Hing Tong showed in 1952 that the most general class of spaces where the theorem holds is the class of perfectly normal spaces. (See Engelking, Exercise 1.7.15(c), p. 62 for details and specific references.)
  2. ^ In particular, there exists   such that   for every natural number  . The necessisty of only considering the tail of   comes from the fact that for small values of   the set   may be empty.

References

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  1. ^ Verry, Matthieu. "Histoire des mathématiques - René Baire".
  2. ^ a b Stromberg, p. 132, Exercise 4
  3. ^ Kurdila, A. J., Zabarankin, M. (2005). "Convex Functional Analysis". Lower Semicontinuous Functionals. Systems & Control: Foundations & Applications (1st ed.). Birkhäuser-Verlag. pp. 205–219. doi:10.1007/3-7643-7357-1_7. ISBN 978-3-7643-2198-7.
  4. ^ Willard, p. 49, problem 7K
  5. ^ Giaquinta, Mariano (2007). Mathematical analysis : linear and metric structures and continuity. Giuseppe Modica (1 ed.). Boston: Birkhäuser. Theorem 11.3, p.396. ISBN 978-0-8176-4514-4. OCLC 213079540.
  6. ^ Puterman, Martin L. (2005). Markov Decision Processes Discrete Stochastic Dynamic Programming. Wiley-Interscience. pp. 602. ISBN 978-0-471-72782-8.
  7. ^ Moore, James C. (1999). Mathematical methods for economic theory. Berlin: Springer. p. 143. ISBN 9783540662358.
  8. ^ "To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous".
  9. ^ Stromberg, p. 132, Exercise 4(g)
  10. ^ "Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions".
  11. ^ a b c d e Freeman, R. A., Kokotović, P. (1996). Robust Nonlinear Control Design. Birkhäuser Boston. doi:10.1007/978-0-8176-4759-9. ISBN 978-0-8176-4758-2..
  12. ^ a b Goebel, R. K. (January 2024). "Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction". Chapter 2: Set convergence and set-valued mappings. Other Titles in Applied Mathematics. Society for Industrial and Applied Mathematics. pp. 21–36. doi:10.1137/1.9781611977981.ch2. ISBN 978-1-61197-797-4.

Bibliography

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