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Continuous function

Definition

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

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Dirichlet's function Dirichlet's test Disc integration Discrete topology Divergence Divergence theorem Domain of a function Domain theory Dover Publications Duality (mathematics)E (mathematical constant)Eduard Heine Encyclopedia of Mathematics Equicontinuity Equivalence of categories Equivalence relation Euler substitution Euler–Maclaurin formula European Mathematical Society Existence theorem Exponential function Exterior derivative Extreme value theorem Factorial Faà di Bruno's formula Filter base Filter on a set Filters in topology Final topology Finite difference First-countable space First derivative test Fluxion Fourier analysis Fourier series Fractional calculus Free variables and bound variables Function (mathematics)Function composition Function of a complex variable Function of a real variable Function of several complex variables Function of several real variables Function space Functional analysis Functor Fundamental Theorem of Line Integrals Fundamental theorem of calculus G-delta set Gabriel's horn General Leibniz rule Generality of algebra Generalizations of the derivative Generalized Stokes' theorem Generalized Stokes theorem Generalized function Geometric calculus Geometric continuity Geometric series Glossary of calculus Gottfried Wilhelm Leibniz Gradient Gradient theorem Graph of a function Green's theorem Gδ set Hahn–Banach theorem Harmonic analysis Harmonic series (mathematics)Hausdorff space Heaviside cover-up method Heaviside step function Heine theorem Helmholtz decomposition Hessian matrix Higher-order function History of calculus History of the function concept Homeomorphism Hypercomplex analysis Hyperreal numbers Hölder continuity Hölder continuous function Identity function Image (mathematics)Implicit differentiation Implicit function Improper integral Indeterminate form Indexed family Indicator function Indiscrete topology Infinite series Infinitesimal Infinitesimal calculus Infinity Initial topology Injective function Integer-valued function Integrable function Integral Integral calculus Integral equation Integral of inverse functions Integral of secant cubed Integral of the secant function Integral test for convergence Integral transform Integration Bee Integration by parts Integration by reduction formulae Integration by substitution Integration using Euler's formula Integro-differential equation Interior (topology)Interior operator Intermediate value theorem Interval (mathematics)Intuition Inverse function Inverse function rule Inverse function theorem Isaac Newton Isolated point Jacobian matrix and determinant James Dugundji Jump discontinuity Karl Weierstrass Kuratowski closure axioms Kuratowski closure operator L'Hôpital's rule Lagrange multiplier Lambda calculus Laplace operator Law of Continuity Least-squares spectral analysis Lebesgue integrability condition Lebesgue integration Lebesgue measurable set Leibniz's notation Leibniz integral rule Leonhard Euler Lim inf Lim sup Limit (category theory)Limit 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topology Microcontinuity Morphism Multiple integral Multiplicative inverse Multivalued function Multivariable calculus Natural logarithm Neighborhood (mathematics)Neighborhood filter Neighborhood system Net (mathematics)Newton's method Newton's notation for differentiation Non-standard analysis Nonstandard analysis Norm (mathematics)Normal function Normed vector space Notation for differentiation Object (category theory)One-sided limit Open and closed maps Open ball Open interval Open map Open set Open subset Order of approximation Order of integration (calculus)Order theory Ordered pair Ordinary differential equation Oscillation (mathematics)P-adic analysis P-adic number Parametric continuity Partial derivative Partial differential equation Partial fractions in integration Partial function Partial ordering Partially ordered set Path-connected Pathological (mathematics)Peter Gustav Lejeune Dirichlet Picard–Lindelöf theorem Piecewise Piecewise function Plain English Pointwise 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functions Springer-Verlag Square root Squeeze mapping Stationary point Steinmetz solid Stepanov-Denjoy theorem Stirling's approximation Stochastic calculus Stochastic differential equation Stokes' theorem Subset Subspace topology Sum rule in differentiation Supremum Surface integral Surjective Surjective function Symmetrically continuous function T0 space Table of derivatives Tangent Tangent function Tangent half-angle substitution Taylor's theorem Taylor series Telescoping series Tensor Tensor calculus Term test The Method of Mechanical Theorems Theorem Thomae's function Tietze extension theorem Topological closure Topological neighborhood Topological space Topology Trapezoidal rule Triangle inequality Trigonometric function Trigonometric substitution Undergraduate Texts in Mathematics Uniform continuity Uniform convergence Uniform convergence theorem Uniform space Uniformly continuous Uniformly continuous function Value (mathematics)Vector calculus Vector calculus identities Vector space Volume integral Weierstrass function Zero of a function Édouard Goursat

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