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Showing 31–60 of 1709 reference entries

Physics · MeasurementCaesium-133 defining frequency

ΔνCs denotes the unperturbed ground-state hyperfine transition frequency of caesium-133. Its numerical value is fixed exactly in SI units.

DefinitionMathematics & logic · InformationClassical propositions and connectives

Classical logic combines truth-valued propositions using defined connectives.

DefinitionMathematics & logicAxioms, deduction and mathematical proof

A mathematical proof derives a conclusion within stated assumptions and rules.

MethodMathematics & logic · InformationSets, membership and basic operations

A set specifies membership without counting duplicate listings.

DefinitionMathematics & logic · InformationFunctions, domain and codomain

A function assigns exactly one output to each input in its domain.

DefinitionMathematics & logic · InformationMathematical induction

A base case and general successor step establish a predicate on all nonnegative integers.

MethodMathematics & logic · MeasurementEuclidean right triangles and the Pythagorean theorem

In a Euclidean right triangle, squared leg lengths sum to the squared hypotenuse length.

Mathematical statementMathematics & logic · PhysicsSine, cosine and the unit circle

Sine and cosine are unit-circle coordinates whose squared values sum to one.

Mathematical statementMathematics & logic · InformationComplex numbers

A complex number has real coordinates a and b and an imaginary unit satisfying i²=−1.

DefinitionMathematics & logic · Information · PhysicsLinear maps and matrix representation

A linear map preserves linear combinations and has a matrix once bases are chosen.

DefinitionMathematics & logicLimit of a real function

A finite limit controls output error through sufficiently small input distance.

DefinitionMathematics & logic · PhysicsDerivative at a point

A derivative is a finite limit of local difference quotients.

DefinitionMathematics & logic · PhysicsDefinite integral and signed accumulation

A definite integral is the limiting signed accumulation of small contributions.

DefinitionMathematics & logic · PhysicsFundamental theorem of calculus

Continuity connects differentiation with integration.

Mathematical statementMathematics & logic · InformationProbability space and axioms

A probability assigns nonnegative, normalised, countably additive weights to events.

DefinitionMathematics & logic · InformationConditional probability and independence

Conditioning restricts attention to an event of positive probability.

DefinitionMathematics & logic · InformationExpected value of a discrete random variable

Expected value weights possible values by their probabilities.

DefinitionMathematics & logic · MeasurementConfidence interval for a population mean

A confidence level describes repeated-sampling coverage of an interval procedure.

MethodInformation · Mathematics & logicBits and unsigned binary encoding

A bit has two possible states; a fixed-width unsigned binary word encodes an integer.

DefinitionInformation · Mathematics & logicShannon entropy of a discrete distribution

Shannon entropy measures uncertainty in a specified probability distribution.

DefinitionInformationAlgorithm and decision procedure

An algorithm specifies effective steps; a decision procedure must terminate on every permitted input.

MethodInformation · Mathematics & logicUndecidability of the halting problem

No total algorithm decides halting for every program and input in a general computation model.

Mathematical statementInformation · Mathematics & logicAsymptotic upper bounds and big O

Big O bounds eventual growth up to a fixed constant factor.

DefinitionMeasurementSI base units

Seven named base units organize the SI quantity system.

DefinitionMeasurementSI decimal prefixes

Prefix symbols specify decimal scale factors.

DefinitionMeasurementMetrological traceability

Connect a result to a reference through documented calibrations.

DefinitionMeasurement · Mathematics & logicFirst-order uncertainty propagation

Combine input uncertainty through a measurement model.

MethodMeasurement · Physics · Mathematics & logicDimensional consistency

Use dimensions to detect impossible quantity equations.

MethodPhysics · Earth & cosmosNewtonian gravitation

An inverse-square force models attraction between point masses.

ModelPhysicsFrequency, wavelength and phase speed

Relate a periodic wave's spatial and temporal repetition.

Definition
DefinitionSource checked

Limit of a real function

Mathematics & logic

A finite limit controls output error through sufficiently small input distance.

Key knowledge

The limit of f(x) as x approaches a is L if every positive output tolerance ε admits a positive input tolerance δ such that 0 < |x−a| < δ implies |f(x)−L| < ε.

Values and equations

limₓ→ₐ f(x) = L

f
Real-valued function
a
Approached input
L
Finite limiting output

Conditions and scope

  • Here f is real-valued and defined on a punctured open interval around a.
  • The value f(a) may differ from L or be undefined; infinite and one-sided limits have separate definitions.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. Calculus Volume 1, 2.5: Precise definition of a limitOpenStaxTextbook chapteropenstax.orgRead source
  2. Cálculo I · Agenda 05: LimitesJean Cerqueira Berni, Universidade de São PauloTeaching resource · PDFime.usp.brRead source

Evidence and review

Finite two-sided epsilon-delta definition.

2026-10-10 · Checked quantifier order, punctured neighbourhood and finite-real scope.

Foundational review

2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.

Sources from other places

Agreement within this scope

Punctured epsilon–delta real finite limits.

The source allows general accumulation-point domains; the repository punctured interval is a valid narrower case.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–4 of 4 relationships

Scientific relationships · 4 / 4

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Uses definition
    Relationship sources · 1

    Linked statements

    • The limit of f(x) as x approaches a is L if every positive output tolerance ε admits a positive input tolerance δ such that 0 < |x−a| < δ implies |f(x)−L| < ε.
  2. Uses definition
    Relationship sources · 1

    Linked statements

    • For a real function defined on an open interval around a, its derivative at a is the limit of [f(a+h)−f(a)]/h as h approaches zero, if that finite limit exists.
  3. Uses definition
    Relationship sources · 1

    Linked statements

    • For a continuous real function on [a,b], the definite integral is the limit of sums f(xᵢ*)Δx over successively finer equal subdivisions. It measures signed accumulation; negative contributions subtract rather than become positive area.
  4. Uses definition
    Relationship sources · 1

    Linked statements

    • An improper integral is the finite limit of proper integrals as a singular endpoint or an infinite endpoint is approached. With multiple singularities or both infinite tails, split the domain and require each one-sided limit separately. Symmetric principal-value cancellation alone does not establish improper convergence.
Entry detailsPermanent link to this entry ↗
Record identity
math-limits
Revision
core.1
Structured reference record
Essential core · first collection