Limit point of a set
Definition
Let be a subset of and let be a real number. The point is a limit point of if for each there is with .
Remarks
A limit point of need not belong to . For example, is a limit point of .
Each point of an interval with more than one point is a limit point of the interval. So are its endpoints.
Limit points of the domain are the points where the limit of a function makes sense.
Mathlib uses the general form AccPt.
Where this leads
363 pages build on this one.
- Sequential characterization of limits
- Power series are differentiable inside the radius of convergence
- A function with zero derivative is constant
- Natural logarithm
- A differentiable function is convex iff its derivative is increasing
- Taylor's theorem with the Cauchy remainder
- Euler–Mascheroni constant
- The logarithm grows slower than any power
- Intermediate value theorem
- Derivative of sine
- A convex function on an open interval is continuous
- 1 + x is at most exp x
- Reduction formula
- Rolle's theorem
- Integral test
- Weierstrass function
- First derivative test
- cos pi = -1
- sin x / x tends to 1
- Tangent line
- Taylor series
- A function with nonnegative derivative is increasing
- Uniformly continuous functions map Cauchy sequences to Cauchy sequences
- Derivative function
- Inverse trigonometric functions
- Power series of sine
- Cauchy mean value theorem
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Taylor's theorem with the Peano remainder
- Absolutely convergent improper integral
- Pythagorean trigonometric identity
- Taylor's theorem
- Indefinite integral
- The Dirichlet function is not Riemann integrable
- Maclaurin series
- log x is at most x - 1
- Inverse hyperbolic functions
- Classification of discontinuities
- Integral of cosine
- Lebesgue's criterion for Riemann integrability
- Jump discontinuity
- Derivative of arctangent
- Harmonic numbers grow like log n
- First fundamental theorem of calculus
- Arctangent
- Integral of the exponential function
- Power series of the exponential function
- Continuous functions are Riemann integrable
- Continuously differentiable function
- Continuous image of a closed interval is a closed interval
- Continuity of the inverse of a strictly monotone continuous function
- Newton's method
- A function with positive derivative is strictly increasing
- Derivatives of the hyperbolic functions
- Derivative of the logarithm
- Differentiable functions are continuous
- Higher derivative
- Infinite limit
- Power series of cosine
- Integration by parts
- One-sided limit
- Smooth non-analytic function
- Uniform continuity on a set
- Differentiable function
- x^2 is not uniformly continuous on the real line
- sin(1/x) has no limit at 0
- Boundedness theorem
- Arcsine
- Mean value theorem
- Smooth function
- Quotient rule
- Antiderivative
- pi
- Polynomial functions are continuous
- Absolute value is continuous but not differentiable at 0
- Uniform convergence of derivatives allows term-by-term differentiation
- Exponential growth beats polynomial growth
- Integration by substitution
- A continuous injective function on an interval is strictly monotone
- The exponential function is convex
- Gaussian integral
- Basel problem
- Trigonometric substitution
- Continuity at a point
- The exponential function is its own derivative
- x sin(1/x) is continuous at 0
- Derivative
- The gamma function extends the factorial
- Comparison test for improper integrals
- Second derivative test
- Derivative of arcsine
- Epsilon-delta definition of a limit
- Critical point
- Quotient of continuous functions is continuous
- Area under a curve
- Integral of sine
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Taylor polynomial
- Thomae's function
- One-sided derivative
- Existence of nth roots by the intermediate value theorem
- Logarithm of a quotient
- Weierstrass approximation theorem
- L'Hôpital's rule
- Power rule
- Lipschitz functions are uniformly continuous
- Integral of 1/x
- Logarithm of a product
- General Leibniz rule
- Stirling's formula
- One-sided limits of a monotone function exist
- The exponential function is strictly increasing
- Derivative of x^a
- Derivative of an inverse function
- Linearity of differentiation
- Derivative of a polynomial
- Mean value theorem for integrals
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Composition of continuous functions is continuous
- Continuous function
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Continuity is equivalent to sequential continuity
- Fermat's theorem on stationary points
- Integral of a power
- Uniform continuity
- Darboux's theorem
- Reciprocal rule
- Removable discontinuity
- Irrationality of pi
- Continuity of the sum of a uniformly convergent series of functions
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- A continuous nowhere differentiable function
- Chain rule
- Power-reduction formula for cosine
- Abel's theorem
- Indeterminate form
- Derivative of tangent
- Power series of arctangent
- Dirichlet function
- Term-by-term differentiation
- Real exponentiation
- Limit of a function
- Derivative of cosine
- Gamma function
- Square root function
- Logarithm of a power
- Product rule
- Improper integral
- Improper integral of x^p on an unbounded interval
- Product of continuous functions is continuous
- Continuous functions on a closed bounded interval are uniformly continuous
- Tangent half-angle substitution
- Epsilon-delta definition of continuity
- Integration by partial fractions
- Rational function
- x^2 sin(1/x) is differentiable with discontinuous derivative
- Taylor's theorem with the integral remainder
- Extreme value theorem
- A monotone function has countably many discontinuities
- Uniform limit theorem
- Wallis product
- Second fundamental theorem of calculus
- Limit at infinity
- Second derivative
- x^n converges pointwise but not uniformly on [0, 1]
- Mercator series
- A function with nonnegative second derivative is convex
- Constant multiple rule
- The logarithm is strictly concave
- Laplace's method
- Beta function in terms of the gamma function
- Arnold tongue
- Stone–Weierstrass theorem
- Arakelyan's theorem
- Lebesgue's theorem on differentiability of monotone functions
- Newton–Kantorovich theorem
- Minimal curvature property of natural cubic splines
- Singular function
- Stein's unbiased risk estimate
- Cauchy–Pompeiu formula
- L2 convergence of Fourier series
- Brianchon's theorem
- Durand–Kerner method
- Regula falsi
- Teichmüller space
- Mehta integral
- Stirling's formula for the gamma function
- Weierstrass product for the gamma function
- Uniformization theorem
- Brent's method
- Mergelyan's theorem
- High indices theorem
- Log-gamma function
- Perron integral
- Runge's theorem
- Bump function
- Stein's lemma
- Bertrand's test
- Gamma function as Mellin transform of the exponential
- Riesz space
- Padé approximant
- Rational approximation
- Fixed-point iteration
- Purely transcendental extension
- Mertens' theorems
- Smoothing spline
- Peano kernel theorem
- Stein's example
- Budan's theorem
- Bisection method
- Reduction of an initial value problem to a Volterra equation
- Holomorphic map between Riemann surfaces
- Richardson extrapolation
- Beta function
- Quasi-analytic function
- Uniform convergence of Bernstein approximations
- Ramanujan's series for 1/pi
- Tauber's theorem
- Khinchin integral
- Conway base 13 function
- Fixed-point property
- Difference quotient
- Continuous functional calculus
- Quadratic convergence of Newton's method
- Symmetric derivative
- Constructive analysis
- Nonlinear regression
- Subderivative
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Modulus of continuity
- Gauss's test
- Mittag-Leffler summation
- Log-convexity of the gamma function
- Baire function
- Lüroth's theorem
- Analytic function between Banach spaces
- Final value theorem
- Mean value theorem for divided differences
- Blancmange curve
- Watson's lemma
- Aberth method
- Tauberian condition
- Landau–Kolmogorov inequality
- Ramanujan summation
- Bairstow's method
- Legendre duplication formula
- Numerical continuation
- Newton fractal
- Smoothness classes
- Uniform convergence of absolutely summable Fourier series
- Gauss–Newton algorithm
- Mahler's theorem
- Karamata's Tauberian theorem
- Differentiation operator as inverse of integration
- Shooting method
- Delta method
- Regular surface
- Laplace transform of the error function
- Logarithmically convex function
- Markov brothers' inequality
- Euler–Boole summation
- Cauchy formula for repeated integration
- f-ring
- Approximation of a kernel by degenerate kernels
- Wirtinger's inequality for functions
- Steffensen's method
- Broyden's method
- Topological field
- Interpolation error formula
- Envelope
- Constructive analysis
- Abscissa of convergence
- Legendre transformation
- Inversion of rational transforms by partial fractions
- Jenkins–Traub algorithm
- Hilbert's seventeenth problem
- Newman's theorem on rational approximation of the absolute value
- Wiman–Valiron theory
- Density of trigonometric polynomials in continuous functions
- Padé table
- Differentiable dependence on parameters
- Spherical function
- Denjoy–Carleman theorem
- Borel's lemma
- Initial value theorem
- Viète's formula
- Heaviside cover-up method
- Polynomial function
- Compactness of integral operators
- Curtis–Hedlund–Lyndon theorem
- Brouwer fixed-point theorem
- Lindelöf summation
- Cantor function
- Commutative Gelfand–Naimark theorem
- Tauberian theorems for integral transforms
- Thin set
- Fourier basis of L2 of the circle
- Indirect method by shooting
- Jentzsch's theorem on positive kernels
- Particular values of the gamma function
- Separation of variables
- Error bound for the trapezoidal rule
- Householder's method
- Korovkin's theorem
- Finite difference coefficients
- Bohr–Mollerup theorem
- Poles of the gamma function
- Sturm's theorem
- Incomplete gamma function
- Volterra's function
- Laplace transform of a power function
- Approximate derivative
- Heun's method
- Halley's method
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Maximum and minimum
- Hölder's theorem
- Euler–Gauss limit formula
- Descartes' rule of signs
- Trigonometric integrals
- Levenberg–Marquardt algorithm
- Fejér's theorem
- Basel problem via Fourier series
- Abel's summation formula
- Ridders' method
- Differential dynamic programming
- Gamma function has no zeros
- Wielandt's theorem
- Reflection formula for the gamma function
- Midpoint method
- Gamma of one half
- Lie symmetry method
- Adjoint state method
- Fubini's theorem on differentiation
- Decay of Fourier coefficients and smoothness
- Disk algebra
- Poncelet's closure theorem
- Degree of exactness of a quadrature rule
- Gauss multiplication formula
- Semi-continuity
- Dini derivative
- Barnes G-function
- Reciprocal gamma function
- Multiplication theorem
- Oscillation of a function
- Lambert summation
- Abel summation
- Chudnovsky algorithm
- Euler integral for the gamma function
- Functional equation of the gamma function
- Holomorphy of the gamma function
- Banach–Zarecki theorem
- Runge's phenomenon
Further reading
- Web https://en.wikipedia.org/wiki/Accumulation_point, Accumulation points of a set
About this page
| Tag | 07EGCNHAVW |
|---|---|
| URI | m2://07EGCNHAVW |
| Aliases | cluster-point-of-a-set |
| Kind | definition |
| Writing | W1 stated |
| MSC | 26A03 |
| Defines | Limit point |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.