Continuous function
Builds on Continuity at a point
Definition
Let and let be a function. The function is continuous if it is continuous at each point of . For a subset , is continuous on if its restriction to is continuous.
Remarks
Sums, products, quotients, and compositions of continuous functions are continuous.
Polynomial functions are continuous. So are the exponential function, the sine, and the cosine.
On a closed bounded interval, a continuous function has the intermediate value property and attains its extreme values.
Mathlib defines it for topological spaces, as Continuous.
Reading path
To understand this page from the ground up, read these 4 pages in order. The foundations come first.
Show the path as a graph
- no evidence level
- Limit point of a set definition
- Limit of a function definition
- Continuity at a point definition
- Continuous function definition
Where this leads
218 pages build on this one.
- 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
- First derivative test
- cos pi = -1
- sin x / x tends to 1
- Taylor series
- A function with nonnegative derivative is increasing
- Inverse trigonometric functions
- Power series of sine
- Cauchy mean value theorem
- Leibniz formula for pi
- Integral of exp(-x) over the half line
- Pythagorean trigonometric identity
- Maclaurin series
- log x is at most x - 1
- Inverse hyperbolic functions
- Integral of cosine
- 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
- A function with positive derivative is strictly increasing
- Derivative of the logarithm
- Power series of cosine
- Integration by parts
- Smooth non-analytic function
- x^2 is not uniformly continuous on the real line
- Boundedness theorem
- Arcsine
- Mean value theorem
- Smooth function
- pi
- Polynomial functions are continuous
- Uniform convergence of derivatives allows term-by-term differentiation
- Integration by substitution
- A continuous injective function on an interval is strictly monotone
- The exponential function is convex
- Basel problem
- Trigonometric substitution
- The gamma function extends the factorial
- Second derivative test
- Derivative of arcsine
- Quotient of continuous functions is continuous
- Area under a curve
- Integral of sine
- Dirichlet integral
- Taylor's theorem with the Lagrange remainder
- Existence of nth roots by the intermediate value theorem
- Logarithm of a quotient
- Weierstrass approximation theorem
- L'Hôpital's rule
- Integral of 1/x
- Logarithm of a product
- Stirling's formula
- The exponential function is strictly increasing
- Derivative of x^a
- Derivative of an inverse function
- Mean value theorem for integrals
- The integral of a continuous function is an antiderivative
- Fundamental theorem of calculus
- Composition of continuous functions is continuous
- Sine is 2π-periodic
- A function with bounded derivative is Lipschitz
- Continuity is equivalent to sequential continuity
- Integral of a power
- Darboux's theorem
- Irrationality of pi
- pi is greater than 3
- Arccosine
- Logarithm is the inverse of the exponential
- Chain rule
- Power-reduction formula for cosine
- Derivative of tangent
- Power series of arctangent
- Term-by-term differentiation
- Real exponentiation
- Derivative of cosine
- Square root function
- Logarithm of a power
- 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
- 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
- Wallis product
- Second fundamental theorem of calculus
- Mercator series
- A function with nonnegative second derivative is convex
- The logarithm is strictly concave
- Arnold tongue
- Stone–Weierstrass theorem
- Arakelyan's theorem
- Lebesgue's theorem on differentiability of monotone functions
- Minimal curvature property of natural cubic splines
- Singular function
- Stein's unbiased risk estimate
- Cauchy–Pompeiu formula
- L2 convergence of Fourier series
- Regula falsi
- Teichmüller space
- Weierstrass product for the gamma function
- Uniformization theorem
- Brent's method
- Mergelyan's theorem
- Runge's theorem
- Bump function
- Stein's lemma
- Bertrand's test
- Riesz space
- Padé approximant
- Rational approximation
- Fixed-point iteration
- Purely transcendental extension
- Mertens' theorems
- 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
- Quasi-analytic function
- Ramanujan's series for 1/pi
- Khinchin integral
- Conway base 13 function
- Fixed-point property
- Difference quotient
- Continuous functional calculus
- Quadratic convergence of Newton's method
- Constructive analysis
- Failure of Aizerman's conjecture
- Frullani integral
- Darboux function
- Gauss's test
- Log-convexity of the gamma function
- Baire function
- Lüroth's theorem
- Analytic function between Banach spaces
- Mean value theorem for divided differences
- Ramanujan summation
- Smoothness classes
- Uniform convergence of absolutely summable Fourier series
- Mahler's theorem
- Differentiation operator as inverse of integration
- Regular surface
- 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
- Topological field
- Interpolation error formula
- Constructive analysis
- Inversion of rational transforms by partial fractions
- 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
- Viète's formula
- Heaviside cover-up method
- Polynomial function
- Compactness of integral operators
- Curtis–Hedlund–Lyndon theorem
- Brouwer fixed-point theorem
- Cantor function
- Commutative Gelfand–Naimark theorem
- Fourier basis of L2 of the circle
- Jentzsch's theorem on positive kernels
- Error bound for the trapezoidal rule
- Bohr–Mollerup theorem
- Sturm's theorem
- Volterra's function
- Noncommutative geometry
- An infinite product converges iff its series of logarithms converges
- Quintic equation
- Maximum and minimum
- Descartes' rule of signs
- Trigonometric integrals
- Fejér's theorem
- Basel problem via Fourier series
- Abel's summation formula
- Ridders' method
- Adjoint state method
- Decay of Fourier coefficients and smoothness
- Disk algebra
- Degree of exactness of a quadrature rule
- Semi-continuity
- Barnes G-function
- Chudnovsky algorithm
- Banach–Zarecki theorem
- Runge's phenomenon
Further reading
About this page
| Tag | 0P27W9RW0G |
|---|---|
| URI | m2://0P27W9RW0G |
| Kind | definition |
| Writing | W1 stated |
| MSC | 26A15 |
| Defines | Continuous function |
| Uses | Continuity at a point |
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What this page needs for W2
- 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.
What links here
Used by
- Intermediate value theorem
- A convex function on an open interval is continuous
- Continuously differentiable function
- x^2 is not uniformly continuous on the real line
- Boundedness theorem
- Polynomial functions are continuous
- Quotient of continuous functions is continuous
- Composition of continuous functions is continuous
- Continuity is equivalent to sequential continuity
- Product of continuous functions is continuous
- Continuous functions on a closed bounded interval are uniformly continuous
- Extreme value theorem
- Arakelyan's theorem
- Singular function
- Mergelyan's theorem
- Riesz space
- Fixed-point iteration
- Holomorphic map between Riemann surfaces
- Khinchin integral
- Conway base 13 function
- Frullani integral
- Darboux function
- Baire function
- Smoothness classes
- Mahler's theorem
- f-ring
- Topological field
- Spherical function
- Polynomial function
- Curtis–Hedlund–Lyndon theorem
- Cantor function
- Fejér's theorem
- Disk algebra
- Semi-continuity
- Banach–Zarecki theorem