In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functionss that is weaker than continuity is upper (respectively, lower) semicontinuous is lower semi-continuous if its epigraph is lower semi-continuous.

then the result is lower semicontinuous.

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

Definitions

is a topological space is a function with values in the extended real numbers.

Upper semicontinuity

is called upper semicontinuous at a point there exists a neighborhood. where lim sup is the limit superior where the infimum.

is a metric space with distance function formulation, similar to the definition of continuous function

is called upper semicontinuous if it satisfies any of the following equivalent conditions:

1. (1) The function is upper semicontinuous at every point of its domain.

2. is open.

3. -superlevel set is closed.

4. (4) The hypograph.

5. is continuous when the codomain is given the left order topology.

Lower semicontinuity

is called lower semicontinuous at a point there exists a neighborhood is the limit inferior

is a metric space with distance function

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

1. (1) The function is lower semicontinuous at every point of its domain.

2. is open.

3. -sublevel set is closed.

4. (4) The epigraph.

5. is continuous when the codomain is given the right order topology.

Examples

piecewise but not lower semicontinuous.

The floor function 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. while the function limits from the left or right at zero do not even exist.

is the space of curves (with the supremum distance its length is lower semicontinuous..

A fundamental example in real analysis is Fatou's lemma is a sequence of non-negative measurable functions denotes the (pointwise) limit inferior denotes the set of positive measurable functions endowed with the topology of convergence in measure is lower semicontinuous.

Properties

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.

• is continuous if and only if it is both upper and lower semicontinuous.

• The characteristic function or indicator function is a closed set is an open set.

• In the field of convex analysis, the characteristic function. With that definition, the characteristic function of any is lower semicontinuous, and the characteristic function of any is upper semicontinuous.

Binary operations on semicontinuous functions

.

• is lower semicontinuous is not the indeterminate form). The same holds for upper semicontinuous functions.

• is lower semicontinuous. The corresponding result holds for upper semicontinuous functions.

• is upper semicontinuous.

• is non-decreasing, then the composition is continuous.

• ) forms a lattice. The corresponding statements also hold for upper semicontinuous functions.

Optimization of semicontinuous functions

• The (pointwise) supremum) is lower semicontinuous.

• In particular, the limit of a monotone increasing

  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.

• is a compact space

• (Proof for the upper semicontinuous case 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

• (Theorem of Baire) be a metric space is the limit of a point-wise increasing such that

• and

• , the continuous functions can be taken to be real-valued.

• is the limit of a monotone decreasing the continuous functions can be taken to be real-valued.

• is locally constant on some dense open subset

  is sequential are sequentially closed. In general, upper semicontinuous functions are sequentially upper semicontinuous, but the converse may be false.

Semicontinuity of set-valued functions

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 The preimage is not disjoint.

Upper and lower semicontinuity

is upper semicontinuous

is lower semicontinuous

in the above definitions with arbitrary topological spaces.

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. is not upper semicontinuous in the set-valued sense.

Inner and outer semicontinuity

is called inner semicontinuous

is called outer semicontinuous).

Hulls

is the lower semicontinuous hull. is defined pointwise by the relation.

The lower semicontinuous hull plays a role in convex analysis. Given a convex (extended real) function, the epigraph might not be closed. But the lower semicontinuous hull of a convex function is convex, and is known as the closure of the original convex function.

Some operations in convex analysis, such as the Legendre transform automatically produce closed convex functions. The Legendre transform applied twice to a convex function gives the closure of the original function, rather than the original function. Thus the lower semicontinuous hull is a way of regularizing convex functions, by modifying it at boundary points of its effective domain.

In categorical terms is the (left) Kan extension is given by the colimit. The upper semicontinuous hull is a right Kan extension.

Other types of hulls are often considered in applications. For example, the infimum of the set of continuous affine functions that majorize a given function on a convex subset of a topological vector space is upper semicontinuous. This fact is used in the proof of the Choquet theorem. Similar ideas applied to subharmonic functions are used in the Perron method for solving the Dirichlet problem for the Laplace operator in a domain. The key condition for the class of subharmonic solutions is upper semicontinuity, particularly near the boundary where the boundary conditions are applied.

Applications

Calculus of variations

An important application of semicontinuity is to the calculus of variations. It derives its significance in this context due to the following theorem..

This is often combined with results such as Tonelli's theorem in functional analysis, which characterizes the weak lower semicontinuity of nonlinear functionals on Lp spaces in terms of the convexity of another function. More specialized results of this kind are useful in variational formulations of problems in partial differential equations, which relate semicontinuity of functionals given by integration to the convexity properties of the integrand, often defined on some Sobolev space. The prototypical example is the Dirichlet problem for the Laplace operator, which can be formulated as a minimization problem of the energy, subject to boundary conditions i.e., the integral of the squared norm of the gradient of a function over a bounded domain in Euclidean space. The integrand is convex in an appropriate Sobolev space, so the limit of a minimizing sequence is a solution of the Dirichlet problem. This has implications, for instance, for finite element solutions, which gives a way to construct a minimizing sequence.

Existence of saddle points

Together with convexity assumptions, both upper and lower semicontinuity play a role in theorems guaranteeing the existence of saddle points of functions, on locally convex topological vector spaces. One such result is the minimax theorem of Fan and Sion. belonging to reflexive Banach spaces, such that

• and

• ,

Dimension

Many integer-valued functions of importance are also semicontinuous. For a simple example, suppose one has a polyhedron is lower semicontinuous. This is intuitively because under any small perturbations, you can move from a face of lower dimension, such as an edge or vertex, to one of higher dimension, but any point of a higher dimensional face cannot be moved to one of lower dimension if the perturbation is small enough.

Another example of a similar character is that matrix rank matrices. This is because the rank can go up at matrices which are nearby, but not down. As a result of this, together with the implicit function theorem, when a Lie group acts smoothly on a smooth manifold).

Algebraic geometry

More sophisticated versions of this same idea play a fundamental role in algebraic geometry, where many dimension maps with codomain in the integers are known to be semicontinuous. (For example as applied to a Newton–Okounkov body.)

be schemes is upper semicontinuous. is coherent can be found in the standard textbook of Hartshorne. Original work in the language of hypercohomology can be found in EGA III Théorème (7.7.5), citing also previous work, in particular Grauert for the complex-analytic setting.

is lower semicontinuous. is upper semicontinuous.

Vakil collected a list of further semicontinuity results in algebraic geometry.

Descriptive set theory

Semicontinuous functions are used in descriptive set theory to define stratifications of topological spaces by complexity measures such as dimension, rank, or ordinal height. Such functions often take values in an ordinal are closed (and hence Borel in a Polish space).

be a tree coded by a point in Baire space. The rank to each tree is lower semicontinuous, analogous to how matrix rank.

More generally, ordinal-valued lower semicontinuous functions are used to measure the complexity of points or structures in a Polish space—such as Scott ranks of countable structures, projective ranks of sets, or Lusin–Novikov complexities of equivalence relations. These functions enable fine classification and are crucial in defining universal setss and effective parametrizations in higher levels of the projective hierarchy.

under a lower semicontinuous function is closed, such functions yield canonical stratifications of topological spaces into closed (thus Borel) pieces of increasing complexity. This property is often used in proofs of reflection principles, separation theorems, and in the effective classification of Borel equivalence relations.

Dynamical systems

In ergodic theory and topological dynamics, semicontinuity arises naturally when studying functionals on the space of invariant measures of a dynamical system. The most important example is the entropy function, which assigns to each invariant measure its measure-theoretic entropy.

under the weak-* topology. The entropy map is an upper semicontinuous function

This property plays a key role in the variational principle over all invariant measures. Upper semicontinuity guarantees that this supremum is attained when the space of measures is compact.

More generally, many functionals of interest—such as Lyapunov exponents, dimension spectra, or return time statistics—are semicontinuous on the space of invariant measures. In some cases, these semicontinuity properties are used to prove existence of measures maximizing or minimizing a given quantity, or to establish structural properties of the simplex—set).

Similar ideas appear in the theory of joinings, where one studies invariant couplings between systems. The set of joinings is compact in the weak-* topology, and semicontinuity is used to analyze disjointness and uniqueness of invariant couplings.

See also

Notes

References

Bibliography