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Semi-continuity

In mathematics, semi-continuity refers to a weakening of the notion of continuity for real-valued functions defined on topological spaces, distinguishing between upper semi-continuity and lower semi-continuity.^[1] A function f: X \to \mathbb{R} is upper semi-continuous at a point x \in X if for every t \in \mathbb{R}, the preimage f^{-1}((-\infty, t)) is open in X, or equivalently, in metric spaces, \limsup_{y \to x} f(y) \leq f(x).^[1] Conversely, f is lower semi-continuous at x if f^{-1}((t, \infty)) is open for all t, or \liminf_{y \to x} f(y) \geq f(x).^[1] A function is continuous if and only if it is both upper and lower semi-continuous.^[2] The concept was first introduced by René Baire in his 1899 doctoral thesis for real-valued functions on \mathbb{R}, using notions of upper and lower limits to address discontinuities in the study of functions of real variables.^[3] Baire's work laid foundational groundwork in real analysis, particularly for understanding pointwise limits and the Baire category theorem, where semi-continuous functions play a role in classifying sets of discontinuity.^[3] Semi-continuity finds broad applications across mathematical fields. In optimization and convex analysis, lower semi-continuous functions ensure the existence of minimizers on compact sets, as they attain their infimum, which is crucial for theorems like Weierstrass's extremum principle.^[1] In the calculus of variations, weak lower semi-continuity of integral functionals guarantees the convergence of minimizing sequences under weak topologies, enabling solutions to problems in partial differential equations and elasticity.^[4] In algebraic geometry, upper semi-continuity describes the behavior of dimensions of fibers or cohomology groups under flat morphisms, as in the semicontinuity theorem for coherent sheaves.^[5] Key properties include closure under pointwise suprema for lower semi-continuous functions (and infima for upper) and preservation under monotone limits: increasing limits of lower semi-continuous functions remain lower semi-continuous.^[2] Examples abound, such as the characteristic function of an open set being lower semi-continuous and that of a closed set being upper semi-continuous, or the arc length functional on paths being lower semi-continuous.^[2] These notions extend to set-valued functions and multifunctions in variational inequalities and game theory, where hemicontinuity (a synonym for semi-continuity) ensures stability of solution sets.^[6]

Definitions

Upper semicontinuity

Upper semicontinuity is a weakening of the notion of continuity for real-valued functions defined on a topological space. Consider a function f: X \to \mathbb{R}, where X is a topological space. The function f is upper semicontinuous at a point x_0 \in X if, for every \varepsilon > 0, there exists a neighborhood U of x_0 such that f(x) \leq f(x_0) + \varepsilon for all x \in U.^[7] This condition ensures that the function values near x_0 do not exceed f(x_0) by more than any prescribed positive amount in some local neighborhood. An equivalent characterization of upper semicontinuity at x_0 relies on the topology of the space. Specifically, f is upper semicontinuous at x_0 if and only if the set \{x \in X \mid f(x) < a\} is open in X for every a \in \mathbb{R}.^[1] This open set formulation highlights that upper semicontinuity preserves openness for subbasic open sets of the form (-\infty, a) in the codomain. A related property is that the \alpha-superlevel sets \{x \in X \mid f(x) \geq \alpha\} are closed in X for every \alpha \in \mathbb{R}, which follows from the sequential condition in metric spaces or the topological closure properties.^[1] In terms of limits, upper semicontinuity at x_0 can be expressed using the limit superior: f(x_0) \geq \limsup_{x \to x_0} f(x).^[1] This means that the function value at x_0 provides an upper bound for the limsup of nearby values, capturing the idea that the function does not "jump up" discontinuously. The function f is said to be globally upper semicontinuous on X if it is upper semicontinuous at every point x \in X.^[7] A continuous function is both upper and lower semicontinuous, but upper semicontinuity alone allows for downward jumps.^[7]

Lower semicontinuity

Lower semicontinuity is a weakening of the notion of continuity for real-valued functions defined on topological spaces, capturing the idea that the function values near a point do not fall below the value at that point by more than any prescribed positive amount. Consider a function f: X \to \mathbb{R}, where X is a topological space. The function f is lower semicontinuous at a point x_0 \in X if, for every \varepsilon > 0, there exists a neighborhood U of x_0 such that f(x) \geq f(x_0) - \varepsilon for all x \in U.^[7] This ensures that f approaches x_0 without sudden decreases in value. An equivalent pointwise condition is that f(x_0) \leq \liminf_{x \to x_0} f(x), or in sequential terms, for every sequence (x_n) in X converging to x_0, f(x_0) \leq \liminf_{n \to \infty} f(x_n).^[8] Globally, f is lower semicontinuous on X if it satisfies the pointwise condition at every x \in X.^[7] Equivalent global characterizations include the openness of the sets \{x \in X \mid f(x) > a\} for all a \in \mathbb{R}, or the closedness of the epigraph \{(x, t) \in X \times \mathbb{R} \mid t \geq f(x)\}.^[7]^[9] The latter implies that the sublevel sets \{x \in X \mid f(x) \leq \alpha\} are closed for every \alpha \in \mathbb{R}.^[9] A continuous function is both lower and upper semicontinuous.^[7]

Examples

Scalar functions

A classic example of a real-valued function that is lower semicontinuous but not upper semicontinuous is the step function defined by f(x) = 0 if x \leq 0 and f(x) = 1 if x > 0. This function is continuous everywhere except at x = 0, where the left-hand limit is 0 and the right-hand limit is 1. At x = 0, the liminf of f(x) as x approaches 0 is 0, which equals f(0), satisfying the lower semicontinuity condition, while the limsup is 1, which exceeds f(0), violating upper semicontinuity. Away from 0, the function is continuous, hence both semicontinuous.^[10] Another illustrative example is the characteristic (or indicator) function of a closed set in \mathbb{R}. For a closed set C \subseteq \mathbb{R}, the function \chi_C(x) = 1 if x \in C and \chi_C(x) = 0 otherwise is upper semicontinuous. This follows because the sublevel sets \{ x \mid \chi_C(x) < a \} are open for all a \in \mathbb{R}: for $0 < a \leq 1, this set is the open complement of C. However, it is generally not lower semicontinuous unless C is also open. For instance, taking C = [0, \infty), \chi_C(x) jumps from 0 to 1 at x = 0 and is upper semicontinuous everywhere, with discontinuity only at the boundary point.^[11] To illustrate a function that is neither upper nor lower semicontinuous, consider a variant of the Dirichlet function defined on \mathbb{R} by d(x) = 1 if x is rational and d(x) = 0 if x is irrational. At every rational point x, where d(x) = 1, the liminf is 0 (approached via irrationals) which is less than 1, so it fails lower semicontinuity, though the limsup is 1 equaling d(x), satisfying upper semicontinuity locally. At every irrational point y, where d(y) = 0, the limsup is 1 (approached via rationals) exceeding 0, failing upper semicontinuity, though the liminf is 0 equaling d(y), satisfying lower semicontinuity locally. Thus, d is neither semicontinuous at any point in the full sense, as the limits oscillate between 0 and 1 everywhere due to the density of rationals and irrationals.^[12] These examples can be visualized through their graphs. The step function appears as a horizontal line at 0 up to and including x = 0, then jumps vertically to 1 for x > 0, with the discontinuity manifesting as an open circle at (0,1) if plotting the right limit, emphasizing the "filled" lower value at the jump. The indicator of a closed interval like [0,1] shows flat lines at 0 outside, jumping to 1 inside, with filled points at the endpoints to reflect upper semicontinuity, where jumps are "downward" from the perspective of approaching the set. The Dirichlet function's graph is dense with points alternating between 0 and 1 in every interval, lacking any consistent jump or limit behavior, resulting in erratic vertical lines at every scale without settling to a semicontinuous profile.

Set-valued functions

Set-valued functions extend the notion of semicontinuity from scalar functions to mappings F: X \rightrightarrows Y between topological spaces, where the image of each point is a nonempty subset of Y. Unlike scalar cases, where inequalities like \limsup f(x) \leq f(x_0) define upper semicontinuity, set-valued variants rely on set inclusions involving limits of sets, such as the Painlevé-Kuratowski upper and lower limits.^[13] A basic example of a set-valued map that is both upper and lower semicontinuous is the constant map F(x) = \{0\} for all x \in \mathbb{R}, where the singleton set remains unchanged regardless of the point, satisfying the inclusion conditions trivially since the limit sets coincide with the image.^[14] To illustrate failure of semicontinuity, consider the correspondence S: \mathbb{R} \rightrightarrows \mathbb{R} defined by S(x) = \{0\} if x is rational and S(x) = \{1\} if x is irrational. This map is neither upper nor lower semicontinuous at any point, as sequences of rationals and irrationals dense in \mathbb{R} lead to limit sets that violate the required inclusions: for instance, approaching any x_0, the upper limit includes both 0 and 1, which cannot be contained in the singleton image.^[13] An example of an upper semicontinuous set-valued map arises from a scalar upper semicontinuous function \phi: X \to \mathbb{R}. Define F(x) = (-\infty, \phi(x)], the closed half-line up to \phi(x). This map is upper semicontinuous, as the decreasing nature of the intervals aligns with the upper limit condition \limsup_{x \to x_0} F(x) \subset F(x_0), equivalent to the closed graph property when values are closed.^[14] Painlevé-Kuratowski convergence underpins these notions, where a sequence of sets C_n converges to C if the upper limit \limsup C_n \subset C and lower limit \liminf C_n \supset C. For a simple illustration, consider C_n = [0, 1 + 1/n] in \mathbb{R}; as n \to \infty, this sequence converges in the Painlevé-Kuratowski sense to [0,1], since points in [0,1] are limits of sequences from all but finitely many C_n, and no points outside [0,1] are limit points of the C_n. This type of convergence is required for sequential characterizations of set-valued semicontinuity at points.^[13]

Properties

Algebraic operations

The sum of two upper semicontinuous functions is upper semicontinuous.^[15] Likewise, the sum of two lower semicontinuous functions is lower semicontinuous.^[15] However, the sum of an upper semicontinuous function and a lower semicontinuous function need not be semicontinuous in either sense. For instance, define f: \mathbb{R} \to \mathbb{R} by f(x) = 0 if x \ge 0 and f(x) = -1 if x < 0; this is upper semicontinuous. Define g: \mathbb{R} \to \mathbb{R} by g(x) = 0 if x \le 0 and g(x) = 1 if x > 0; this is lower semicontinuous. Their sum h(x) = f(x) + g(x) equals -1 for x < 0, $0atx = 0

, and &#36;1

for x > 0. At x = 0, sequences approaching from the right yield \limsup h(x_n) = 1 > h(0), so h is not upper semicontinuous, while sequences from the left yield \liminf h(x_n) = -1 < h(0), so h is not lower semicontinuous. The pointwise infimum of an arbitrary family of upper semicontinuous functions is upper semicontinuous, since \{x \mid \inf_\alpha f_\alpha(x) < a\} = \bigcup_\alpha \{x \mid f_\alpha(x) < a\} is a union of open sets and hence open.^[1] Dually, the pointwise supremum of an arbitrary family of lower semicontinuous functions is lower semicontinuous.^[16] For composition, if f is upper semicontinuous, g is continuous and non-decreasing, then g \circ f is upper semicontinuous, because \limsup (g \circ f)(x_n) \le g(\limsup f(x_n)) \le g(f(x)) by continuity and monotonicity of g.^[15] A similar preservation holds for lower semicontinuous f with non-decreasing continuous g. The product of two upper semicontinuous functions is upper semicontinuous if both are non-negative, as multiplication is continuous and jointly non-decreasing on [0, \infty) \times [0, \infty).^[3] Without the positivity assumption, the product need not preserve upper semicontinuity.

Optimization principles

In optimization theory, lower semicontinuity plays a crucial role in guaranteeing the existence of minima for functions defined on compact sets. Specifically, a variant of the Weierstrass extreme value theorem states that if f: K \to \mathbb{R} is lower semicontinuous and K is a nonempty compact topological space, then f attains its minimum value on K.^[17] This result weakens the continuity assumption of the classical Weierstrass theorem while preserving the attainment of the infimum, as the closed sublevel sets \{x \in K : f(x) \leq c\} for each c \in \mathbb{R} ensure that minimizing sequences have convergent subsequences with limits in the domain.^[18] Dually, upper semicontinuity ensures the existence of maxima on compact sets. If f: K \to \mathbb{R} is upper semicontinuous on a nonempty compact space K, then f attains its maximum value on K, since the superlevel sets \{x \in K : f(x) \geq c\} are closed and thus compact, allowing the supremum to be achieved.^[1] This principle is fundamental in problems where one seeks to maximize objectives under compactness constraints, such as in finite-dimensional programming or topological optimization. Lower semicontinuity also manifests through the geometry of the epigraph, defined as \operatorname{epi} f = \{(x, t) \in X \times \mathbb{R} : f(x) \leq t\}. A proper extended real-valued function f is lower semicontinuous if and only if its epigraph is a closed subset of the product topology.^[19] For convex functions, this closedness equates to lower semicontinuity, ensuring that the closure of the epigraph remains convex and provides a closed convex envelope that preserves minimization properties.^[20] In the direct method of the calculus of variations, lower semicontinuity is essential for proving the existence of minimizers of integral functionals. The approach involves constructing a minimizing sequence in a reflexive Banach space, extracting a weakly convergent subsequence via compactness (e.g., via coercivity and weak lower semicontinuity), and applying lower semicontinuity to pass to the limit, yielding \liminf I(u_n) \geq I(u) where u is the weak limit and I is the functional, thus attaining the infimum. This technique, rooted in the work of Tonelli and refined in modern treatments, relies on the functional's lower semicontinuity with respect to weak convergence to ensure the inequality holds.^[21]

Topological characteristics

A function f: X \to \mathbb{R} defined on a topological space X is continuous at a point x_0 \in X if and only if it is both upper semicontinuous and lower semicontinuous at x_0.^[7] This equivalence holds because continuity requires that the limit exists and equals the function value, which aligns precisely with the limsup condition for upper semicontinuity (\limsup_{x \to x_0} f(x) \leq f(x_0)) and the liminf condition for lower semicontinuity (\liminf_{x \to x_0} f(x) \geq f(x_0)), ensuring \lim_{x \to x_0} f(x) = f(x_0).^[7] Semicontinuous functions belong to Baire class 1, meaning they can be expressed as pointwise limits of sequences of continuous functions. Specifically, every upper or lower semicontinuous function on \mathbb{R} (or more generally on a metric space) arises as such a limit, distinguishing them from higher Baire classes while encompassing all continuous functions as a subclass. This property underscores their role in the Baire hierarchy, where pointwise limits preserve certain regularity but allow for discontinuities of the first kind. Upper semicontinuous functions are Borel measurable, as the superlevel sets \{x \mid f(x) \geq a\} are closed for every a \in \mathbb{R}, generating the Borel \sigma-algebra.^[22] Similarly, lower semicontinuous functions are Borel measurable because the sublevel sets \{x \mid f(x) \leq a\} are closed for every a \in \mathbb{R}.^[22] This measurability facilitates their integration in measure theory, contrasting with more irregular functions like the Dirichlet function. Unlike continuous functions, which satisfy the intermediate value property (Darboux property) on connected domains, semicontinuous functions generally lack this trait.^[7] For instance, the Heaviside step function H(x) = 0 if x < 0 and H(x) = 1 if x \geq 0 is upper semicontinuous on \mathbb{R} but fails the Darboux property on [-1, 1], as it maps to \{0, 1\} without attaining values in (0, 1), despite connecting points where H(-1) = 0 and H(1) = 1.^[7] This absence highlights how semicontinuity permits jump discontinuities that violate connectedness in the image.

Set-Valued Semicontinuity

Upper and lower variants

In the context of set-valued functions F: X \rightrightarrows Y, where X and Y are topological spaces, upper semicontinuity at a point x_0 \in X is defined as follows: for every open set V \subset Y such that F(x_0) \subset V, there exists a neighborhood U of x_0 in X with the property that F(x) \subset V for all x \in U. This condition ensures that the images under F near x_0 remain contained within any open superset of F(x_0), analogous to the scalar case where the function values do not exceed those at x_0 in the limit. Lower semicontinuity at x_0 requires that for every y_0 \in F(x_0) and every open set V \subset Y containing y_0, there exists a neighborhood U of x_0 such that F(x) \cap V \neq \emptyset for all x \in U. This guarantees that points near x_0 map to sets intersecting any open neighborhood of elements in F(x_0), mirroring the scalar notion where function values approach from above. A set-valued function is continuous at x_0 if it is both upper and lower semicontinuous there. When the values of F are closed subsets of Y and X is a metric space, upper semicontinuity at x_0 is equivalent to the graph of F being closed in a neighborhood of (x_0, F(x_0)), assuming local boundedness of F. This closed graph property provides a useful sequential characterization: if sequences x_n \to x_0 and y_n \in F(x_n) with y_n \to y, then y \in F(x_0). These notions extend naturally via Painlevé-Kuratowski convergence of sets, where the limit superior \limsup_{x \to x_0} F(x) consists of all limit points of sequences y_n \in F(x_n) with x_n \to x_0, and the limit inferior \liminf_{x \to x_0} F(x) consists of points y such that for every open neighborhood V of y, there exists a neighborhood U of x_0 with F(x) \cap V \neq \emptyset for all x \in U. Upper semicontinuity holds if \limsup_{x \to x_0} F(x) \subset F(x_0), lower semicontinuity if F(x_0) \subset \liminf_{x \to x_0} F(x), and full continuity if equality holds in both inclusions. This convergence framework unifies the topological conditions in non-Hausdorff or general settings.

Inner and outer variants

In the context of set-valued functions defined on metric spaces, inner and outer semicontinuity provide sequential characterizations that offer an alternative perspective to the topological definitions of upper and lower semicontinuity, emphasizing limit behaviors along sequences.^[23] These notions are particularly useful when analyzing mappings with potentially unbounded or non-compact ranges, where topological conditions may impose stricter requirements.^[23] Outer semicontinuity of a set-valued map F: X \rightrightarrows Y at a point x \in X is defined sequentially as follows: for every sequence x_n \to x in X and every sequence y_n \in F(x_n) such that y_n \to y in Y, it holds that y \in F(x).^[23] This condition is equivalent to \limsup_{n \to \infty} F(x_n) \subset F(x), where the limit superior captures all possible limit points of sequences from the images.^[23] Notably, outer semicontinuity coincides with the closedness of the graph of F at (x, y) for y \in F(x), making it robust even for mappings without compact values.^[23] Inner semicontinuity at x requires that for every sequence x_n \to x and every y \in F(x), there exists a sequence y_n \in F(x_n) with y_n \to y.^[23] Equivalently, F(x) \subset \liminf_{n \to \infty} F(x_n), where the limit inferior consists of points approachable by sequences from the images.^[23] This ensures that points in the image at the limit point remain attainable nearby, distinguishing it from outer semicontinuity by focusing on inclusion in the opposite direction.^[23] A set-valued map that is both inner and outer semicontinuous at x is hemicontinuous there, and if it is single-valued at x with local boundedness, this implies full continuity in the standard sense.^[23] These properties together yield \limsup_{n \to \infty} F(x_n) \subset F(x) \subset \liminf_{n \to \infty} F(x_n), ensuring the images converge setwise along sequences.^[23] Examples illustrate where inner and outer semicontinuity hold while standard upper or lower semicontinuity may fail due to non-compact ranges. Consider F: \mathbb{R} \rightrightarrows \mathbb{R} defined by F(t) = [0, 1/|t|] for t \neq 0 and F(0) = [0, \infty); this map is outer semicontinuous at $0 because the graph is closed, capturing all limit points from unbounded sequences, but it fails standard upper semicontinuity without compactness assumptions.^[23] Conversely, for compact ranges like F(t) = [0, 1] constantly, both inner and outer semicontinuity hold trivially, aligning with standard notions but without requiring boundedness elsewhere.^[23] In non-compact cases, such as proximal mappings in optimization, outer semicontinuity persists despite unbounded images, providing stability where topological variants do not.^[23]

Envelopes and Hulls

Semicontinuous envelopes

In mathematical analysis, the upper semicontinuous envelope of a function f: X \to \overline{\mathbb{R}}, where X is a topological space and \overline{\mathbb{R}} = [-\infty, \infty], is defined pointwise as

f^*(x) = \inf_{U \ni x} \sup_{y \in U} f(y),

with the infimum taken over all neighborhoods U of x. This construction yields the smallest upper semicontinuous function that dominates f pointwise, meaning f^*(x) \geq f(x) for all x \in X, and any other upper semicontinuous majorant g of f satisfies g \geq f^*.^[24] Dually, the lower semicontinuous envelope is given by

f_*(x) = \sup_{U \ni x} \inf_{y \in U} f(y),

the supremum over neighborhoods U of x. This produces the largest lower semicontinuous function dominated by f, so f_*(x) \leq f(x) for all x \in X, and for any lower semicontinuous minorant h of f, it holds that h \leq f_*.^[24] These envelopes satisfy f_*(x) \leq f(x) \leq f^*(x) at every point x, with equality holding throughout the domain if and only if f is semicontinuous (specifically, lower semicontinuous for equality with f_* and upper semicontinuous for equality with f^*). In uniform spaces, the envelopes provide pointwise approximations to f, with the difference f^* - f_* measuring the oscillatory behavior of f and vanishing uniformly on compact subsets if f is uniformly continuous.

Semicontinuous hulls

The lower semicontinuous convex hull of a proper extended real-valued function f: X \to \overline{\mathbb{R}}, where X is a Banach space, is defined as the pointwise supremum of all lower semicontinuous convex functions that minorize f, denoted \cl \co f or f^{**}. This hull coincides with f itself if and only if f is already lower semicontinuous and convex.^[25] It provides the tightest convex lower semicontinuous relaxation of f, ensuring the epigraph of the hull is the closed convex hull of the epigraph of f.^[26] One standard construction of the lower semicontinuous convex hull employs the Legendre-Fenchel biconjugate, where the convex conjugate (or Fenchel transform) of f is f^*(y) = \sup_{x \in X} (\langle y, x \rangle - f(x)) for y \in X^*, and the biconjugate is

f^{**}(x) = \sup_{y \in X^*} (\langle y, x \rangle - f^*(y)).

This operation yields a proper lower semicontinuous convex function, and f^{**} \leq f pointwise, with equality holding on the domain of f under the stated conditions.^[25] An alternative construction uses iterated infimal convolutions: the inf-convolution of f with a quadratic \frac{1}{2\lambda} \|\cdot\|^2 for \lambda > 0, repeated and scaled as \lambda \to 0, converges pointwise to the hull, preserving lower semicontinuity and convexity at each step.^[27] For set-valued maps f: X \rightrightarrows Y with convex closed values in a finite-dimensional normed space Y, the upper semicontinuous hull, denoted \usc f, is the smallest upper semicontinuous convex-valued map majorizing f. It is constructed via the Kuratowski-Painlevé upper limit:

(\usc f)(x) = \limsup_{z \to x} f(z) = \bigcap_{\delta > 0} \cl \co \left( \bigcup_{z \in B_\delta(x)} f(z) \right),

where \cl \co denotes closed convex hull and B_\delta(x) is the open ball of radius \delta around x.^[28] This hull closes the graph of f under upper hemicontinuity operations, ensuring \gr(\usc f) \supset \cl (\gr f) and preserving Cesàro upper semicontinuity (property (Q)) for multifunctions.^[28] The support function characterization confirms upper semicontinuity: \sigma_{\usc f(\cdot)}(y^*) is upper semicontinuous at x for all y^* \in \ri((0^+ f(x))^\circ).^[28] Preservation theorems guarantee that semicontinuity is maintained under convex hull operations in compact settings. For an upper semicontinuous function f defined on a compact convex subset of a topological vector space, the convex hull \co f—the pointwise supremum of affine minorants—is upper semicontinuous.^[29] More generally, if \{f_\alpha\} is a family of upper semicontinuous functions uniformly bounded above on a compact set, their convex hull is upper semicontinuous, relying on the μ-compactness of the domain to control measure supports in barycentric representations.^[29] For set-valued maps, the closed convex hull of the images under a compact upper hemicontinuous map remains upper hemicontinuous. In optimization and regularization, semicontinuous hulls smooth non-semicontinuous objectives by replacing them with convex lower semicontinuous surrogates that preserve essential solution properties. The argmin set of f relates to that of its lower semicontinuous convex hull via asymptotic cones and recession directions, ensuring minimizers of the hull lie in the closure of convex combinations of near-minimizers of f.^[26] This facilitates stable numerical approximations in nonconvex problems, such as variational inequalities, where the hull enables Fenchel subdifferential calculus without losing optimality conditions.^[26] Unlike non-convex semicontinuous envelopes, these hulls enforce convexity to promote global regularity in objective smoothing.^[26]

Applications

Calculus of variations

In the calculus of variations, lower semicontinuity is essential for establishing the existence of minimizers in the direct method, which constructs a minimizing sequence in a suitable function space and extracts a weak limit point that preserves or lowers the functional values.^[30] Specifically, for an integral functional I(u) = \int_\Omega f(x, u(x), \nabla u(x)) \, dx defined on a reflexive Banach space like W^{1,p}(\Omega) with $1 < p < \infty, weak lower semicontinuity of I combined with coercivity—ensuring that sublevel sets are bounded—and weak convergence of minimizing sequences imply that the sequence converges weakly to a minimizer of I. This framework guarantees attainment of the infimum provided the sublevel sets are weakly compact, a property derived from the reflexivity of the space and the Palais-Smale condition in some cases.^[30] Tonelli's theorem provides a foundational result for weak lower semicontinuity of such functionals when the integrand satisfies Carathéodory conditions—measurable in x, continuous in the other variables—and convexity in the gradient argument. Precisely, if f: \Omega \times \mathbb{R}^N \times \mathbb{R}^{nN} \to [0, \infty) is a Carathéodory function that is convex and continuous in the last variable p, bounded below by an integrable function, and satisfies superlinear growth at infinity (e.g., c(|p|^p - 1) \leq f(x, z, p) \leq C(|p|^p + 1) for some $1 < p < \infty), then I is sequentially weakly lower semicontinuous in W^{1,p}(\Omega; \mathbb{R}^N).^[30] This convexity ensures that the integral functional respects weak limits via Jensen's inequality applied to the convex integrand, facilitating the passage to the limit in the direct method.^[31] Examples illustrate the consequences of semicontinuity failure, particularly in minimal surface problems where the area functional may not admit smooth minimizers without relaxation techniques. In the Plateau problem, seeking surfaces of least area spanning a given boundary curve, the non-quasiconvexity of certain integrands modeling prescribed mean curvature leads to infima not achieved in W^{1,1} or smooth classes, as minimizing sequences develop oscillations or concentrations that lower the energy below any candidate limit. Growth conditions significantly influence the preservation of lower semicontinuity, with quadratic growth enabling straightforward applications in Hilbert spaces like [W](/page/W)^{1,2}. For integrands satisfying c(1 + |\nabla u|^2) \leq f(x, u, \nabla u) \leq C(1 + |\nabla u|^2), convexity or quasiconvexity suffices for weak lower semicontinuity, as the bounded second-order terms align with the Hilbert structure and Poincaré inequalities for coercivity.^[32] In contrast, superlinear growth, such as |\nabla u|^p with p > 2, demands stronger conditions like polyconvexity to counteract the higher-order nonlinearity, which can amplify oscillations in minimizing sequences and require relaxation to the convex hull for existence in spaces like [W](/page/W)^{1,p}.^[33]

Game theory and saddle points

In game theory, semicontinuity is essential for proving the existence of saddle points in zero-sum games and Nash equilibria in more general strategic interactions, often by weakening the continuity assumptions of classical theorems like von Neumann's minimax theorem. A saddle point for a payoff function f(x, y) occurs at (x^*, y^*) where f(x^*, y) \leq f(x^*, y^*) \leq f(x, y^*) for all x \in X and y \in Y, representing a value that the minimizer cannot exceed and the maximizer cannot fall below. Semicontinuity ensures such points exist under conditions where full continuity fails, particularly when payoffs exhibit quasiconvexity or quasiconcavity, allowing the application of fixed-point arguments on convex compact strategy sets. Sion's minimax theorem extends von Neumann's result to quasiconvex-quasiconcave functions f: X \times Y \to \mathbb{R}, where X and Y are convex compact subsets of topological vector spaces, f(\cdot, y) is lower semicontinuous and quasiconvex for each y \in Y, and f(x, \cdot) is upper semicontinuous and quasiconcave for each x \in X; under these conditions, \min_{x \in X} \max_{y \in Y} f(x, y) = \max_{y \in Y} \min_{x \in X} f(x, y), guaranteeing a saddle value and optimal strategies. In zero-sum games, this replaces the joint continuity requirement of von Neumann's theorem (1928), as semicontinuity suffices to control the behavior of upper and lower level sets, ensuring the minimax equality holds even for discontinuous payoffs like those in certain pursuit-evasion models. For instance, in a zero-sum game with quasiconvex costs for the minimizer and quasiconcave rewards for the maximizer on compact convex sets, lower semicontinuity in the minimizer's variable prevents the value from "jumping down" unexpectedly, while upper semicontinuity in the maximizer's variable bounds upward deviations.^[34] In set-valued games, where payoffs are multivalued correspondences (e.g., due to uncertainty or multiple outcomes), upper semicontinuity of the payoff correspondence—meaning the inverse images of open sets containing graph points are open—combined with nonempty compact convex values, ensures the existence of Nash equilibria via fixed-point theorems like Kakutani's. Here, a Nash equilibrium is a strategy profile where no player's best response correspondence, upper semicontinuous under these assumptions, deviates profitably; this applies to games with incomplete information, where the correspondence maps joint strategies to sets of possible payoffs. Berge's maximum theorem complements this by establishing continuity of the value function v(\theta) = \max_{x \in \Gamma(\theta)} f(x, \theta) and upper semicontinuity of the argmax correspondence \Gamma(\theta), assuming f is continuous in (x, \theta) and \Gamma is upper semicontinuous with nonempty compact values; in games, this continuity propagates through best-response dynamics to prove equilibrium existence in parameterized settings like Bayesian games.

Geometric measure theory

In geometric measure theory, the concept of lower semicontinuity plays a pivotal role in establishing the existence of minimizers for variational problems involving currents and varifolds, particularly through the analysis of the mass functional. For integral currents, the mass functional, defined as the total variation norm of the current, exhibits lower semicontinuity with respect to weak convergence in the space of currents with bounded mass and boundary mass. This property ensures that if a sequence of integral currents converges weakly to a limit current, the mass of the limit is no greater than the liminf of the masses of the sequence, facilitating the direct method in the calculus of variations.^[35] Compactness theorems further leverage this lower semicontinuity to guarantee the existence of limiting objects in sequences with controlled mass. The Federer-Fleming compactness theorem states that any sequence of integral currents in a Euclidean space with uniformly bounded mass and boundary mass contains a weakly convergent subsequence to another integral current, with the mass of the limit controlled by the bounds on the sequence. For varifolds, a similar result is provided by Allard's compactness theorem, which asserts that a sequence of integral varifolds with uniformly bounded mass and first variation converges, after passing to a subsequence, in the weak-* topology to an integral varifold, preserving key geometric properties like stationarity. Almgren's foundational work on the structure of stationary varifolds complements these by providing higher-dimensional regularity and compactness estimates that rely on semicontinuous energy functionals. A canonical application arises in the Plateau problem, where one seeks a minimal area surface spanning a given boundary curve. The lower semicontinuity of the mass functional and the Federer-Fleming compactness theorem imply the existence of an integral current minimizer with the prescribed boundary, as sequences of approximating surfaces with controlled area converge to a limit of no larger mass. This resolves the classical problem in higher codimensions, where smooth spanning surfaces may not exist, by allowing singular minimizers. Recent developments since 2000 have extended these ideas to semicontinuous perimeter functionals in more general settings, such as those defined via measure data, yielding new isoperimetric inequalities. For instance, lower semicontinuity results for weighted perimeters on sets of finite perimeter ensure the existence of isoperimetric minimizers under relaxed boundary conditions, with applications to nonlocal functionals and quantitative stability estimates.^[36] These advancements refine classical isoperimetric profiles by incorporating semicontinuous relaxations that handle non-smooth data while preserving compactness.^[37]

Algebraic geometry

In algebraic geometry, semicontinuity theorems provide essential tools for analyzing how geometric invariants behave in families of schemes or varieties parametrized over a base space. These results are particularly vital for proper or projective morphisms, where they describe the variation of fiber dimensions, cohomology dimensions, and related polynomials across the parameter space. Such semicontinuity ensures that loci where these invariants attain certain values form constructible or open sets, facilitating the study of moduli spaces and degenerations. A fundamental result is the upper semicontinuity of fiber dimensions under proper morphisms. For a proper morphism f: X \to Y of schemes, the function y \mapsto \dim X_y is upper semicontinuous, meaning that for each integer k, the set \{ y \in Y \mid \dim X_y \leq k \} is open in Y. This follows from the fact that the set where the dimension exceeds k is the image under f of a closed subset of X, which remains closed due to properness. Although sometimes associated with Grauert in the analytic setting, the algebraic version appears in the Éléments de géométrie algébrique (EGA) and is a cornerstone for understanding the geometry of fibrations. In the projective case, this implies upper semicontinuity of the degree of the Hilbert polynomial of the fibers, as the degree equals the relative dimension. Cohomology groups exhibit similar behavior. For a proper morphism f: X \to Y of noetherian schemes and a coherent sheaf \mathcal{F} on X that is flat over Y, the function y \mapsto \dim_k H^i(X_y, \mathcal{F}_y) is upper semicontinuous for each i \geq 0. Known as the semicontinuity theorem, this result underpins Grauert's theorem, which asserts that if Y is reduced and the dimension of H^i(X_y, \mathcal{F}_y) is constant on a connected component of Y, then the higher direct image sheaf R^i f_* \mathcal{F} is locally free on that component. These theorems apply to families of coherent sheaves on varieties, such as line bundles, where the loci of constant cohomology rank are open. The Hilbert polynomial also displays controlled variation in families. In flat projective morphisms, the Hilbert polynomial of the structure sheaf on the fibers is constant across the base, reflecting the preservation of Euler characteristics under base change. In more general (non-flat) settings, particularly arithmetic ones involving local rings or completions over discrete valuation rings, the coefficients of the Hilbert-Samuel polynomial exhibit lower semicontinuity with respect to flat deformations of ideals, ensuring that multiplicities do not decrease abruptly in special fibers. For instance, the Hilbert-Samuel multiplicity function is lower semicontinuous in flat families of ideals in local rings. In deformation theory, these semicontinuity properties manifest in the behavior of tangent and obstruction spaces. For a smooth variety X, the dimension of H^1(X, T_X) governs the local dimension of the moduli space of deformations, and upper semicontinuity implies that this dimension can jump upward at singular points in the parameter space, signaling the emergence of singularities in deformed fibers. Conversely, drops in cohomology dimensions may indicate resolutions or smoothing, with jumps highlighting non-smoothness in the moduli.

Descriptive set theory

In Polish spaces, upper semicontinuous functions are Borel measurable, as they belong to Baire class 1, being pointwise limits of continuous functions. Consequently, the image under an upper semicontinuous function of any Borel subset of a Polish space is an analytic set, a fundamental property known as the Souslin property in this context. This follows from the general fact that Borel images of Borel sets under Borel measurable maps between Polish spaces are analytic. Similarly, lower semicontinuous functions share this measurability and thus exhibit the analogous property for their images. Semicontinuous functions also interact with the Baire category structure in Polish spaces. As Baire class 1 functions, they map comeager sets to comeager sets under certain conditions, preserving non-meagerness in a way that aligns with the Baire category theorem; for instance, the image of a comeager set under a Baire class 1 function is comeager in the range if the function is surjective onto a Baire space. This preservation aids in analyzing the category of sets defined via semicontinuous maps, ensuring that meager sets do not map to non-meager sets in complete metric spaces. A representative example arises with set-valued upper semicontinuous maps, where the graph is a Borel set in the product Polish space. The projection of this graph onto the codomain yields a Suslin (analytic) set, illustrating how semicontinuity generates sets of controlled descriptive complexity from Borel data. For real-valued upper semicontinuous functions on Polish spaces, the graph itself is Borel, and its projection onto the range recovers the image as an analytic set, reinforcing the role of semicontinuity in bounding set-theoretic hierarchies. In connections to determinacy, semicontinuity appears in the analysis of Gale-Stewart games with Borel payoff sets, where the value function—indicating the winning player—exhibits semicontinuous behavior under parameterizations of the payoff. For Borel games, Martin's theorem establishes determinacy, and semicontinuous perturbations of payoff sets maintain Borel complexity, ensuring the game's value remains descriptively simple and determined. This links semicontinuity to the broader framework of Borel determinacy, where such functions help characterize winning strategies without increasing the descriptive level beyond analytic.

Dynamical systems

In dynamical systems, upper semicontinuity of attractors is a fundamental property that describes how attractors behave under small perturbations of the system parameters. For families of continuous flows parameterized by a variable, such as in ordinary differential equations, the attractor corresponding to a perturbed parameter lies within an arbitrarily small neighborhood of the original attractor when the perturbation is sufficiently small. This ensures that the limiting behavior of trajectories remains robust, with the Hausdorff distance between attractors approaching zero as parameters converge. Such upper semicontinuity holds under conditions like compactness of the phase space and continuity of the flow with respect to parameters, as established in foundational results for non-autonomous and random dynamical systems.^[38] Lower semicontinuity plays a complementary role in stability analysis through Lyapunov functions, which quantify the rate of convergence to stable sets. In perturbed systems, a lower semicontinuous Lyapunov function—meaning its value at a limit point is no greater than the limit of values along approaching sequences—guarantees that stable and attractive sets persist under small disturbances. This property allows the function to bound the energy or distance to an attractor from below, preventing sudden expansions that could destabilize the system, and is particularly useful for proving asymptotic stability in infinite-dimensional settings like partial differential equations. Generalized theorems extend this to non-smooth cases, where lower semicontinuous functions certify both local and global stability without requiring differentiability.^[39]^[40] Semicontinuity properties with respect to parameter variations aid in understanding transitions to chaotic dynamics. In the 2020s, applications of semicontinuity have advanced control theory, particularly for designing robust attractors in systems subject to uncertainties. For instance, in controlled piezoelectric beams with nonlinear boundary feedback, upper semicontinuity of pullback attractors ensures that stabilization persists despite parameter drifts, allowing controllers to maintain dissipative behavior and bounded long-term dynamics. Similarly, in non-autonomous stochastic systems with delays, lower semicontinuity of Lyapunov-like functions supports the robustness of attractors under external forcing, enabling reliable control strategies for applications like vibration suppression in engineering structures. These developments highlight semicontinuity's role in certifying the resilience of controlled dynamical systems to perturbations.^[41]^[42]

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