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{{short description|Property of functions which is weaker than continuity}}{{For|the notion of upper or lower semi-continuous set-valued function|Hemicontinuity}}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 f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than fleft(x_0right).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 x_0 to fleft(x_0right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to fleft(x_0right) - c then the result is lower semicontinuous.(File:Upper semi.svg|thumb|right|An upper semicontinuous function that is not lower semicontinuous. The solid blue dot indicates fleft(x_0right).) (File:Lower semi.svg|thumb|right|A lower semicontinuous function that is not upper semicontinuous. The solid blue dot indicates fleft(x_0right).)The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899.WEB, Verry, Matthieu, Histoire des mathématiques - René Baire,weblink

Definitions

Assume throughout that X is a topological space and f:Xtooverline{R} is a function with values in the extended real numbers overline{R}=R cup {-infty,infty} = [-infty,infty].

Upper semicontinuity

A function f:Xtooverline{R} is called upper semicontinuous at a point x_0 in X if for every real y > fleft(x_0right) there exists a neighborhood U of x_0 such that f(x)y}.

Examples

Consider the function f, piecewise defined by:f(x) = begin{cases}-1 & mbox{if } x < 0, end{cases}This function is upper semicontinuous at x_0 = 0, but not lower semicontinuous.The floor function f(x) = lfloor x rfloor, which returns the greatest integer less than or equal to a given real number x, is everywhere upper semicontinuous. Similarly, the ceiling function f(x) = lceil x rceil 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.Willard, p. 49, problem 7K For example the functionf(x) = begin{cases}sin(1/x) & mbox{if } x neq 0,1 & mbox{if } x = 0,end{cases}is upper semicontinuous at x = 0 while the function limits from the left or right at zero do not even exist.If X = R^n is a Euclidean space (or more generally, a metric space) and Gamma = C([0,1], X) is the space of curves in X (with the supremum distance d_Gamma(alpha,beta) = sup{d_X(alpha(t),beta(t)):tin[0,1]}), then the length functional L : Gamma to [0, +infty], which assigns to each curve alpha its length L(alpha), is lower semicontinuous.BOOK, Giaquinta, Mariano,weblink Mathematical analysis : linear and metric structures and continuity, 2007, Birkhäuser, Giuseppe Modica, 978-0-8176-4514-4, 1, Boston, Theorem 11.3, p.396, 213079540, 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 sqrt 2.Let (X,mu) be a measure space and let L^+(X,mu) denote the set of positive measurable functions endowed with thetopology of convergence in measure with respect to mu. Then by Fatou's lemma the integral, seen as an operator from L^+(X,mu) to [-infty, +infty] is lower semicontinuous.

Properties

Unless specified otherwise, all functions below are from a topological space X to the extended real numbers overline{R}= [-infty,infty]. 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 f:Xtooverline{R} is continuous if and only if it is both upper and lower semicontinuous.
• The indicator function of a set Asubset X (defined by mathbf{1}_A(x)=1 if xin A and 0 if xnotin A) is upper semicontinuous if and only if A is a closed set. It is lower semicontinuous if and only if A is an open set.

In the context of convex analysis, the characteristic function of a set A is defined differently, as chi_{A}(x)=0 if xin A and chi_A(x) = infty if xnotin A. With that definition, the characteristic function of any {{em|closed set}} is lower semicontinuous, and the characteristic function of any {{em|open set}} is upper semicontinuous.

• The sum f+g of two lower semicontinuous functions is lower semicontinuousBOOK, Puterman, Martin L., Markov Decision Processes Discrete Stochastic Dynamic Programming,weblink limited, 2005, Wiley-Interscience, 978-0-471-72782-8, 602, (provided the sum is well-defined, i.e., f(x)+g(x) is not the indeterminate form -infty+infty). The same holds for upper semicontinuous functions.
• If both functions are non-negative, the product function f g of two lower semicontinuous functions is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.
• A function f:Xtooverline{R} is lower semicontinuous if and only if -f is upper semicontinuous.
• The composition f circ g of upper semicontinuous functions is not necessarily upper semicontinuous, but if f is also non-decreasing, then f circ g is upper semicontinuous.BOOK, Moore, James C., Mathematical methods for economic theory,weblink limited, 1999, Springer, Berlin, 9783540662358, 143,
• The minimum and the maximum of two lower semicontinuous functions are lower semicontinuous. In other words, the set of all lower semicontinuous functions from X to overline{R} (or to R) forms a lattice. The same holds for upper semicontinuous functions.
• The (pointwise) supremum of an arbitrary family (f_i)_{iin I} of lower semicontinuous functions f_i:Xtooverline{R} (defined by f(x)=sup{f_i(x):iin I}) is lower semicontinuous.WEB, To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous,weblink

• (Theorem of Baire)The result was proved by René Baire in 1904 for real-valued function defined on R. 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.) Assume X is a metric space. Every lower semicontinuous function f:Xtooverline{R} is the limit of a monotone increasing sequence of extended real-valued continuous functions on X; if f does not take the value -infty, the continuous functions can be taken to be real-valued.Stromberg, p. 132, Exercise 4(g)WEB, Show that lower semicontinuous function is the supremum of an increasing sequence of continuous functions,weblink

• If C is a compact space (for instance a closed bounded interval [a, b]) and f : C to overline{R} is upper semicontinuous, then f has a maximum on C. If f is lower semicontinuous on C, it has a minimum on C.

• Any upper semicontinuous function f : X to N on an arbitrary topological space X is locally constant on some dense open subset of X.
• 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.

See also

• {{annotated link|left-continuous|Directional continuity}}
• {{annotated link|KatÄ›tov–Tong insertion theorem}}
• {{annotated link|Hemicontinuity|Semicontinuous set-valued function}}

Notes

{{reflist|group=note}}

References

{{reflist}}

Bibliography

• JOURNAL, Benesova, B., Kruzik, M., 2017, Weak Lower Semicontinuity of Integral Functionals and Applications, 10.1137/16M1060947, SIAM Review, 59, 4, 703–766, 1601.00390, 119668631,
• BOOK, Bourbaki, Nicolas, Elements of Mathematics: General Topology, 1–4, Springer, 1998, 0-201-00636-7

,

• BOOK, Bourbaki, Nicolas, Elements of Mathematics: General Topology, 5–10, Springer, 1998, 3-540-64563-2

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• BOOK, Engelking, Ryszard, Ryszard Engelking

  publisher=Heldermann Verlag, Berlin

  isbn=3-88538-006-4, BOOK, Gelbaum, Bernard R., Olmsted, John M.H., Counterexamples in analysis, Dover Publications, 2003, 0-486-42875-3 , BOOK, Hyers

  author2=Isac, George, Rassias, Themistocles M., Topics in nonlinear analysis & applications, World Scientific, 1997, 981-02-2534-2, BOOK , Stromberg , Karl , Introduction to Classical Real Analysis , Wadsworth , 1981 , 978-0-534-98012-2 , {{Willard General Topology}} {{Zălinescu Convex Analysis in General Vector Spaces 2002}} {{Convex analysis and variational analysis}}

  
  
  
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