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Topics referred to by the same term
the continuity theorem may refer to one of the following results: the Lévy continuity theorem on random variables; the Kolmogorov continuity theorem on
Continuity_theorem
Form of continuity for functions
absolute continuity is a regularity property of functions that is stronger than uniform continuity, and hence continuity. The notion of absolute continuity allows
Absolute_continuity
Mathematical theorem
In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints on the moments
Kolmogorov_continuity_theorem
Result in probability theory
In probability theory, Lévy’s continuity theorem, or Lévy's convergence theorem, named after the French mathematician Paul Lévy, connects convergence
Lévy's_continuity_theorem
modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process
Lévy's modulus of continuity theorem
Lévy's_modulus_of_continuity_theorem
Mathematical function with no sudden changes
Picard–Lindelöf theorem concerning the solutions of ordinary differential equations. Another, more abstract, notion of continuity is the continuity of functions
Continuous_function
Strong form of uniform continuity
theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness
Lipschitz_continuity
Topics referred to by the same term
results: Lévy's continuity theorem, on random variables Kolmogorov continuity theorem, on stochastic processes In geometry: Parametric continuity, for parametrised
Continuity
Theorem in mathematics
In calculus and real analysis, the mean value theorem (or Lagrange's mean value theorem) is a theorem about differentiable functions, roughly stating
Mean_value_theorem
Mathematical theorem
for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations
Symmetry of second derivatives
Symmetry_of_second_derivatives
Continuous real function on a closed interval has a maximum and a minimum
point in the domain. Use continuity to show that the image of the subsequence converges to the supremum. Boundedness Theorem—If f(x) is continuous on
Extreme_value_theorem
French mathematician (1886-1971)
Cramér's decomposition theorem Lévy distribution Lévy metric Lévy's modulus of continuity Lévy–Prokhorov metric Lévy's continuity theorem Lévy's zero-one law
Paul_Lévy_(mathematician)
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
Relationship between derivatives and integrals
predate the fundamental theorem of calculus by hundreds of years; for example, in the fourteenth century the notions of continuity of functions and motion
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
(mathematical series) Le Cam's theorem (probability theory) Lévy continuity theorem (probability) Lévy's modulus of continuity theorem (probability) Martingale
List_of_theorems
Calculus of functions of several variables
is embodied by the integral theorems of vector calculus: Gradient theorem Stokes' theorem Divergence theorem Green's theorem. In a more advanced study of
Multivariable_calculus
Topics referred to by the same term
Glaeser's theorem may refer to: Glaeser's composition theorem Glaeser's continuity theorem This disambiguation page lists mathematics articles associated
Glaeser's_theorem
Continuous function on an interval takes on every value between its values at the ends
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b] and s is
Intermediate_value_theorem
Uniform restraint of the change in functions
(ordinary) continuity there is a locally applicable δ that depends on both ε and x. So uniform continuity is a stronger continuity condition than continuity; a
Uniform_continuity
Semicontinuity for set-valued functions
Ancel–Granas–Górniewicz–Kryszewski theorem). The upper and lower hemicontinuity might be viewed as usual continuity: Theorem— A set-valued function Γ : A ⇉
Hemicontinuity
Mathematical concept in measure theory
approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem states: Stepanov-Denjoy
Approximately continuous function
Approximately_continuous_function
Topics referred to by the same term
representation theorem In probability theory Hahn–Kolmogorov theorem Kolmogorov extension theorem Kolmogorov continuity theorem Kolmogorov's three-series theorem Kolmogorov's
Kolmogorov's_theorem
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Every polynomial has a real or complex root
The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Existence and uniqueness of solutions to initial value problems
known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard,
Picard–Lindelöf_theorem
Fourier transform of the probability density function
variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Characterizes the continuity of the derivative of the square roots of C2 functions
In mathematical analysis, Glaeser's continuity theorem is a characterization of the continuity of the derivative of the square roots of functions of class
Glaeser's_continuity_theorem
Equation describing the transport of some quantity
ways that the net rate Σ may be altered. By the divergence theorem, a general continuity equation can also be written in a "differential form": ∂ρ∂t+∇⋅𝐣=σ
Continuity_equation
Bound on eigenvalues
In mathematics, the Gershgorin circle theorem (also called Gershgorin Disk Theorem) may be used to bound the spectrum of a square matrix. It was first
Gershgorin_circle_theorem
Theorem in physics showing the conservation of energy for the electromagnetic field
The theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation. Poynting's theorem states
Poynting's_theorem
Statement relating differentiable symmetries to conserved quantities
of a physical quantity is usually expressed as a continuity equation. The formal proof of the theorem utilizes the condition of invariance to derive an
Noether's_theorem
Theorem in calculus relating line and double integrals
we are in position to prove the theorem: Proof of Theorem. Let ε be an arbitrary positive real number. By continuity of A, B and compactness of R‾, given
Green's_theorem
Theorem in projective geometry
upon projection to another plane. Degenerate conics follow by continuity (the theorem is true for non-degenerate conics, and thus holds in the limit
Pascal's_theorem
Function in mathematical analysis
mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function
Modulus_of_continuity
Lévy continuity theorem Darmois–Skitovich theorem Edgeworth series Helly–Bray theorem Kac–Bernstein theorem Location parameter Maxwell's theorem Moment-generating
List_of_probability_topics
Averages of repeated trials converge to the expected value
function of the constant random variable μ, and hence by the Lévy continuity theorem, X‾n converges in distribution to μ: X‾n→𝒟μforn→∞. μ is a constant
Law_of_large_numbers
Provides conditions for a parametric optimization problem to have continuous solutions
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement
Maximum_theorem
Any individual whose preferences satisfy four axioms has a utility function
transitivity, continuity, and independence. These axioms, apart from continuity, are often justified using the Dutch book theorems (whereas continuity is used
Von Neumann–Morgenstern utility theorem
Von_Neumann–Morgenstern_utility_theorem
German mathematician (1815–1897)
definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter
Karl_Weierstrass
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
the continuity question. Asymptotic distribution Big O in probability notation Skorokhod's representation theorem The Tweedie convergence theorem Slutsky's
Convergence of random variables
Convergence_of_random_variables
Approximation of a function by a polynomial
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the
Taylor's_theorem
Method for finding limits in calculus
calculus, the squeeze theorem (also known as the sandwich theorem, the two policemen and a drunk theorem among other names) is a theorem regarding the limit
Squeeze_theorem
Theorem in complex analysis
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard
Cauchy's_integral_theorem
French mathematician (1918–2002)
mathematical education and introduced Glaeser's composition theorem and Glaeser's continuity theorem. Glaeser was a Ph.D. student of Laurent Schwartz. On 3
Georges_Glaeser
Key result in Hamiltonian mechanics and statistical mechanics
n-dimensional divergence theorem. The proof is based on the fact that the evolution of ρ obeys an 2n-dimensional version of the continuity equation: ∂ρ∂t+∇→⋅(ρu→)=0
Liouville's theorem (Hamiltonian)
Liouville's_theorem_(Hamiltonian)
Theorem in measure theory
Egorov's theorem states that pointwise convergence is nearly uniform, and uniform convergence preserves continuity. The strength of Lusin's theorem might
Lusin's_theorem
Theorem which asserts the existence of an object
should be defined in terms of "local uniform continuity". One could get another explanation of existence theorem from type theory, in which a proof of an
Existence_theorem
Limit type in multivariable calculus
\lim _{x\to a}f(x)=L=\lim _{n\to \infty }L_{n}} . A corollary is the continuity theorem for uniform convergence as follows: Corollary 7.1. If lim n → ∞ f
Iterated_limit
functional analysis, Namioka's theorem is a result concerning the relationship between separate continuity and joint continuity of functions defined on product
Namioka's_theorem
Theorems on the convergence of bounded monotonic sequences
In real analysis, the monotone convergence theorem is any of a number of related theorems proving, under certain conditions, the convergence of monotonic
Monotone_convergence_theorem
Special form of continuity
similar to local Dini continuity implies convergence of a Fourier transform. Kellogg's theorem Stenflo, Örjan (2001). "A note on a theorem of Karlin". Statistics
Dini_continuity
Consistent set of finite-dimensional distributions will define a stochastic process
extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees
Kolmogorov_extension_theorem
Soviet mathematician (1903–1987)
Kolmogorov–Arnold theorem Kolmogorov–Arnold–Moser theorem Kolmogorov continuity theorem Kolmogorov's criterion Kolmogorov extension theorem Kolmogorov's three-series
Andrey_Kolmogorov
Mathematical rule for inverting probabilities
Bayes' theorem determines the posterior distribution from the prior distribution. Uniqueness requires continuity assumptions. Bayes' theorem can be generalized
Bayes'_theorem
Conditions for switching order of integration in calculus
Fubini's theorem gives the conditions under which a double integral can be computed as an iterated integral, i.e. by integrating in one variable at a
Fubini's_theorem
Theorem in vector calculus
Stokes' theorem, also known as the Kelvin–Stokes theorem, is a theorem in vector calculus that relates the behavior of a vector field along the edge of
Stokes'_theorem
Concept in statistics
uniformly distributed random phase. Where applicable, the central limit theorem dictates that at any point, the sum of these individual plane-wave contributions
Gaussian_random_field
Property of functions which is weaker than continuity
analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function
Semi-continuity
Representation of a type of random process
then Xt is also a Gaussian process. In other cases, the central limit theorem indicates that Xt will be approximately normally distributed when φ is
Autoregressive_model
Theorem in mathematics
In mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that
Inverse_function_theorem
Statement about integration on manifolds
generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about
Generalized_Stokes_theorem
Probability theorem
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random
Continuous_mapping_theorem
uncertainty Kolmogorov backward equation Kolmogorov continuity theorem Kolmogorov extension theorem Kolmogorov's criterion Kolmogorov's generalized criterion
List_of_statistics_articles
Swedish logician, philosopher, and mathematical statistician
(2): 119–170. doi:10.2307/2986734. JSTOR 2986734. Martin-Löf, P. The continuity theorem on a locally compact group. Teor. Verojatnost. i Primenen. 10 1965
Per_Martin-Löf
Theorem regarding the existence of a solution to a differential equation
Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees
Peano_existence_theorem
Mathematical theorem in real analysis
under the uniform norm. The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and
Uniform_limit_theorem
Mathematical theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is locally
Rademacher's_theorem
Theorems connecting continuity to closure of graphs
closed graph theorem is a result connecting the continuity of a linear operator to a topological property of their graph. Precisely, the theorem states that
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Theorem in topology
fixed-point theorem. Because the properties involved (continuity, being a fixed point) are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent
Brouwer_fixed-point_theorem
Mathematical rule for evaluating limits
L'Hôpital's rule (/ˌloʊpiːˈtɑːl/ loh-pee-TAHL) is a mathematical theorem used for evaluating the limit of a quotient of two functions, each of which tends
L'Hôpital's_rule
In the study of zero sum games, Glicksberg's theorem (also Glicksberg's existence theorem) is a result that shows certain games have a minimax value. If
Glicksberg's_theorem
Stochastic volatility model used in derivatives markets
stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem
SABR_volatility_model
On converting relations to functions of several real variables
In multivariable calculus, the implicit function theorem is a theorem that provides sufficient conditions under which a planar curve specified by F(x,y)=0
Implicit_function_theorem
French mathematician (1789–1857)
thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity. In a paper published
Augustin-Louis_Cauchy
Modern application of infinitesimals
versatile spectrum of foundational approaches, including a definition of continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an
Nonstandard_calculus
intuitionistic derivation of the fan theorem, a key result used in the derivation of the uniform continuity theorem. It is also useful in giving constructive
Bar_induction
Mathematical theorem related to real and functional analysis
mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another
Kirszbraun_theorem
Expressing a measure as an integral of another
In mathematics, the Radon–Nikodym theorem, named after Johann Radon and Otto M. Nikodym, is a result in measure theory that expresses the relationship
Radon–Nikodym_theorem
Theorem on extension of bounded linear functionals
the theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem and has numerous uses in convex geometry. The theorem is
Hahn–Banach_theorem
Solution to a stochastic differential equation
and local martingale property. Uniqueness follows from the Lipschitz continuity of σ,b. In fact, La;b+∂∂s coincides with the infinitesimal generator 𝒜
Diffusion_process
Mathematical concept
equivalent to Definition A when the underlying measure space is finite (see Theorem 2 below), Definition H is widely adopted in Mathematics. The following
Uniform_integrability
Theorem in measure theory
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures
Prokhorov's_theorem
Differentiation under the integral sign formula
For t fixed, the mean value theorem implies there exists z in the interval [x, x + δ] such that fδ(x,t)=fx(z,t). Continuity of fx(x, t) and compactness
Leibniz_integral_rule
Branch of statistics mathematics
machinery can be subsequently applied. Continuity of sample paths can be shown using Kolmogorov continuity theorem. Functional data are considered as realizations
Functional_data_analysis
Inequality on approximations of a function by algebraic or trigonometric polynomials
by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its derivatives. Informally
Jackson's_inequality
Mode of convergence of a function sequence
defined. The following result states that continuity is preserved by uniform convergence: Uniform limit theorem—Suppose E is a topological space, M is a
Uniform_convergence
Sufficient criterion for uniform convergence
In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space
Dini's_theorem
Operation in calculus
this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integration to differentiation and provides
Integral
Any real function on R admits a continuous restriction on a dense subset of R
behaved (with respect to continuity) that all of its restrictions to all possible dense subsets are discontinuous. The theorem's conclusion becomes more
Blumberg_theorem
Foundational law of electromagnetism relating electric field and charge distributions
as Gauss's flux theorem or sometimes Gauss's theorem, is one of Maxwell's equations. It is an application of the divergence theorem, and it relates the
Gauss's_law
Closed graph theorem – Theorem relating continuity to graphs Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of
Borel_graph_theorem
Textbook by Augustin-Louis Cauchy (1821)
23-25, Cauchy presents eight theorems on properties of infinitesimals of various orders. This section is entitled "Continuity of functions". Cauchy writes:
Cours_d'analyse
3D generalization of the Leibniz integral rule
calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or simply the Reynolds theorem, named after Osborne Reynolds
Reynolds_transport_theorem
Mathematical theorem about functions
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively
Fourier_inversion_theorem
Theorem in mathematics and economics
absolute continuity, differentiability almost everywhere, or left- and right-differentiability, suffice. In particular, Milgrom and Segal's (2002) Theorem 2
Envelope_theorem
Number of intersection points of algebraic curves and hypersurfaces
Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form the theorem states that
Bézout's_theorem
Mathematics of real numbers and real functions
Compactness plays an important role through its interaction with continuity. The extreme value theorem in calculus, for example, says that a continuous function
Real_analysis
Degree of differentiability of a function or map
results such as the Paley–Wiener theorem. Conversely, decay of the Fourier transform can imply differentiability or continuity properties of the original function
Smoothness
equations. It is thus a generalization of a theorem by Édouard Goursat, which instead of assuming the continuity of f, assumes its Fréchet differentiability
Looman–Menchoff_theorem
Branch of mathematics
curves. These two branches are related to each other by the fundamental theorem of calculus. Calculus uses convergence of infinite sequences and infinite
Calculus
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