Semi-continuity

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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.

An upper semicontinuous function that is not lower semicontinuous. The solid blue dot indicates
A lower semicontinuous function that is not upper semicontinuous. 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]

DefinitionsEdit

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

Upper semicontinuityEdit

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  .

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) All sets   with   are open in  , where  .
(3) All superlevel sets   with   are 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 semicontinuityEdit

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  .

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) All sets   with   are open in  , where  .
(3) All sublevel sets   with   are closed in  .
(4) The epigraph   is closed in  .
(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  .

ExamplesEdit

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.[3] 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.

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.

PropertiesEdit

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 from semicontinuity over the whole domain.

  • A function   is continuous if and only if it is both upper and lower semicontinuous.
  • The 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.[note 1]
  • The sum   of two lower semicontinuous functions is lower semicontinuous[4] (provided the sum is well-defined, i.e.,   is not the indeterminate form  ). The same holds for upper semicontinuous functions.
  • If both functions are non-negative, the product function   of two lower semicontinuous functions is lower semicontinuous. The same holds for upper semicontinuous functions.
  • A function   is lower semicontinuous if and only if   is upper semicontinuous.
  • The composition   of upper semicontinuous functions is not necessarily upper semicontinuous, but if   is also non-decreasing, then   is upper semicontinuous.[5]
  • The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from   to   (or to  ) forms a lattice. The same holds for upper semicontinuous functions.
  • The (pointwise) supremum of an arbitrary family   of lower semicontinuous functions   (defined by  ) is lower semicontinuous.[6]
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.
  • (Theorem of Baire)[note 2] Assume   is a metric space. Every lower semicontinuous function   is the limit of a monotone increasing sequence of extended real-valued continuous functions on  ; if   does not take the value  , the continuous functions can be taken to be real-valued.[7][8]
And 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.
  • If   is a compact space (for instance a closed bounded interval  ) and   is upper semicontinuous, then   has a maximum on  . If   is lower semicontinuous on  , it has 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.)
  • Any upper semicontinuous function   on an arbitrary topological space   is locally constant on some dense open subset of  

See alsoEdit

NotesEdit

  1. ^ In the context 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.
  2. ^ 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.)

ReferencesEdit

  1. ^ Verry, Matthieu. "Histoire des mathématiques - René Baire".
  2. ^ a b Stromberg, p. 132, Exercise 4
  3. ^ Willard, p. 49, problem 7K
  4. ^ Puterman, Martin L. (2005). Markov Decision Processes Discrete Stochastic Dynamic Programming. Wiley-Interscience. pp. 602. ISBN 978-0-471-72782-8.
  5. ^ Moore, James C. (1999). Mathematical methods for economic theory. Berlin: Springer. p. 143. ISBN 9783540662358.
  6. ^ "To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous".
  7. ^ Stromberg, p. 132, Exercise 4(g)
  8. ^ "Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions".

BibliographyEdit