Real functions
701 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 201 of its 212 ideas have a page so far.
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- AxiomCompleteness axiom
Let be a nonempty subset of the real numbers that is bounded above. Then has a supremum in .
- PropositionArchimedean property
Let be an Archimedean ordered field and let be an element of . Then there is a natural number with .
- ConceptDedekind cut
A Dedekind cut splits the rational numbers into a lower part and an upper part. Each real number corresponds to exactly one cut, so the cuts give a construction of the real numbers from the rational numbers.
- ConceptCauchy sequence
In a Cauchy sequence, any two terms with large enough indices are as near to each other as we wish. A metric space is complete when each Cauchy sequence in it converges, and the real numbers form a complete space. The node Cauchy sequence gives the precise definition.
- TheoremCompleteness of the real numbers
Let be an Archimedean ordered field. Then the following five conditions on are equivalent.
- TheoremMonotone convergence theorem for sequences
Let be an increasing real sequence, in the sense of a monotone sequence. Assume that is bounded above. Then converges, and its limit is the supremum .
Map of the subject
The real line 13 of 15
- real number definition
- Completeness axiom statement
- Archimedean property statement
- extended real number line definition
- Dedekind cut definition
- Cauchy sequence definition
- Completeness of the real numbers statement
- Monotone convergence theorem for sequences statement
- Bolzano–Weierstrass theorem statement
- Heine–Borel theorem statement
- Nested intervals theorem statement
- Limit superior definition
- Squeeze theorem statement
- The real numbers are uncountable statement
- Cantor set object
Limits and continuity 18 of 22
- Limit of a function definition
- One-sided limit definition
- continuous function definition
- Classification of discontinuities concept
- Uniform continuity definition
- Lipschitz condition definition
- Hölder continuity definition
- Modulus of continuity definition
- Semi-continuity definition
- Oscillation of a function definition
- intermediate value theorem statement
- extreme value theorem statement
- Heine–Cantor theorem statement
- Dirichlet function example
- Thomae's function example
- Weierstrass function counterexample
- Cantor function counterexample
- Volterra's function counterexample
- Blancmange curve example
- Conway base 13 function counterexample
- Baire function definition
- Darboux function definition
Sequences and series of functions 12 of 14
- Pointwise convergence definition
- Uniform convergence definition
- Uniform limit theorem statement
- Weierstrass M-test statement
- Dini's theorem statement
- Arzelà–Ascoli theorem statement
- Equicontinuity definition
- Term-by-term differentiation statement
- Weierstrass approximation theorem statement
- Bernstein polynomial definition
- power series definition
- radius of convergence definition
- Abel's theorem statement
- Tauber's theorem statement
Differentiation 31 of 32
- Derivative definition
- Differentiable function definition
- Difference quotient definition
- One-sided derivative definition
- Dini derivative definition
- Symmetric derivative definition
- product rule statement
- Quotient rule statement
- Chain rule statement
- Derivative of an inverse function statement
- Fermat's theorem on stationary points statement
- Rolle's theorem statement
- Mean value theorem statement
- Cauchy mean value theorem statement
- Darboux's theorem statement
- L'Hôpital's rule statement
- Higher derivative definition
- Smoothness classes definition
- Taylor's theorem statement
- Taylor series definition
- Analytic function definition
- Smooth non-analytic function counterexample
- Bump function definition
- Borel's lemma statement
- Quasi-analytic function definition
- Denjoy–Carleman theorem statement
- Critical point definition
- Maximum and minimum definition
- First derivative test statement
- Second derivative test statement
- Inflection point definition
- Newton's method concept
Monotone functions, bounded variation, absolute continuity 13 of 13
- Monotone function definition
- Lebesgue's theorem on differentiability of monotone functions statement
- Fubini's theorem on differentiation statement
- Bounded variation definition
- Total variation definition
- Jordan decomposition of a function of bounded variation statement
- Absolute continuity definition
- Luzin N property definition
- Banach–Zarecki theorem statement
- Singular function definition
- Minkowski question-mark function example
- Rademacher's theorem statement
- Approximate derivative definition
Elementary and special functions of a real variable 11 of 12
- exponential function definition
- Natural logarithm definition
- Trigonometric functions definition
- Inverse trigonometric functions definition
- Hyperbolic functions definition
- Real exponentiation definition
- Gamma function definition
- Bohr–Mollerup theorem statement
- Beta function definition
- Stirling's formula statement
- Wallis product statement
- Euler–Maclaurin formula statement
Convex functions 11 of 11
- Convex function definition
- Concave function definition
- Strictly convex function definition
- A convex function on an open interval is continuous statement
- A function with nonnegative second derivative is convex statement
- Logarithmically convex function definition
- Quasiconvex function definition
- Subderivative definition
- Legendre transformation definition
- Schur-convex function definition
- Karamata's inequality statement
Classical inequalities 21 of 21
- Inequality of arithmetic and geometric means statement
- Generalized mean inequality statement
- Jensen's inequality statement
- Cauchy–Schwarz inequality statement
- Hölder's inequality statement
- Minkowski inequality statement
- Young's inequality for products statement
- Bernoulli's inequality statement
- Rearrangement inequality statement
- Chebyshev's sum inequality statement
- Hardy's inequality statement
- Hilbert's inequality statement
- Carleman's inequality statement
- Hermite–Hadamard inequality statement
- Muirhead's inequality statement
- Grönwall's inequality statement
- Wirtinger's inequality for functions statement
- Landau–Kolmogorov inequality statement
- Nesbitt's inequality statement
- Schur's inequality statement
- Majorization definition
Real polynomials and rational functions 11 of 11
- Polynomial function definition
- Descartes' rule of signs statement
- Sturm's theorem statement
- Budan's theorem statement
- Gauss–Lucas theorem statement
- Markov brothers' inequality statement
- Bernstein's inequality for polynomials statement
- Chebyshev polynomials object
- Lagrange polynomial definition
- Hilbert's seventeenth problem problem
- Positivstellensatz statement
Functions of several real variables 26 of 27
- Partial derivative definition
- Directional derivative definition
- Total derivative definition
- Jacobian matrix definition
- Gradient definition
- Hessian matrix definition
- Symmetry of second derivatives statement
- Taylor's theorem in several variables statement
- Inverse function theorem statement
- Implicit function theorem statement
- Lagrange multipliers statement
- Multiple integral definition
- Change of variables in multiple integrals statement
- Line integral of a vector field definition
- Surface integral definition
- Green's theorem statement
- Divergence theorem statement
- Stokes' theorem statement
- Generalised Stokes theorem statement
- Differential form definition
- Poincaré lemma for one-forms on convex sets statement
- vector calculus identities concept
- Sard's theorem statement
- Kolmogorov–Arnold representation theorem statement
- Whitney extension theorem statement
- Kirszbraun theorem statement
- McShane–Whitney extension statement
Further topics 14 of 14
- Fractional calculus concept
- Riemann–Liouville integral definition
- Caputo fractional derivative definition
- Cauchy formula for repeated integration statement
- Grünwald–Letnikov derivative definition
- Absolutely monotonic function definition
- Bernstein's theorem on monotone functions statement
- Operator monotone function definition
- Löwner's theorem statement
- Cauchy's functional equation problem
- Pompeiu derivative example
- q-derivative definition
- Time-scale calculus concept
- Interval arithmetic concept
More ideas in this subject
Concepts 147
- Construction of the reals from Cauchy sequences
- Even and odd functions
- Local maximum
- Multivalued function
- Partial fraction decomposition
- Partition of an interval
- Periodic function
- Strictly monotone function
- Arc length
- Littlewood polynomial
- Local minimum
- Mesh of a partition
- Refinement of a partition
- Schur stable polynomial
- Slowly varying function
- Step function
- Abel equation
- Abel function
- Cobweb plot
- Completely monotone function
- Epigraph
- Function of several variables
- Functional square root
- Hadamard finite part
- Homogeneity equation
- Indeterminate form
- Infinite limit
- Iterated function
- Iterative root
- Jensen's functional equation
- Lebesgue integral
- Limit at infinity
- Regular variation
- Stieltjes function
- Vector-valued function
- Continuous function
- Convex function on R^n
- Critical point
- Double integral
- Integrable function
- Iterated derivative
- Iterated integral
- Iterative functional equation
- Jump discontinuity
- Level set
- Limit of a function of several variables
- Lipschitz class
- Local maximum and minimum
- Loeb measure
- Mittag-Leffler function
- Modulus of smoothness
- Removable discontinuity
- Riemann integrable function
- Riesz potential
- Sequential continuity
- Space of functions of bounded variation
- Strict differentiability
- Subadditive function
- Superadditive function
- Tangent line
- Vector field
- Analytic function
- Clarke generalized gradient
- Continuity in R^n
- Derivative function
- Differentiable function of several variables
- Fenchel conjugate
- Higher partial derivatives
- Indefinite integral
- Lebesgue–Stieltjes measure
- P-adic analysis
- Quasi-arithmetic mean
- Rectifiable curve
- Sine and cosine
- Standard part function
- Triple integral
- Upper and lower Darboux sums
- Centre of mass
- Contour line
- Convex integrand
- Definite integral
- Flux through a surface
- Fractional differential equation
- Hemicontinuity
- Improper multiple integral
- Moment of inertia
- Monad in nonstandard topology
- Rank-one convexity
- Semicontinuity
- Square root
- Uniform continuity on a set
- Upper and lower Darboux integrals
- Volterra equations with completely monotone kernels
- Curl
- Divergence
- Germ of a smooth function
- Integration over a non-rectangular region
- Jacobian determinant
- Limit along different paths
- Line integral of a scalar field
- Morrey quasiconvexity
- Multi-index notation
- Second derivative
- Square root function
- Symmetric decreasing rearrangement
- Tangent plane to a graph
- Work done by a force field
- Caccioppoli set
- Dynamic equation on time scales
- Fractional iteration
- Hyperreal number
- Linear approximation
- Littlewood's three principles
- Order on the reals
- Polyconvexity
- Scalar potential
- Absolute value
- Continuously differentiable function
- Fractional integration
- Interval
- Saddle point
- Smooth function
- Conservative vector field
- Laplace–Stieltjes transform
- Perimeter of a set
- Del operator
- Fractional Laplacian
- Floor function
- Ceiling function
- Nonstandard analysis
- Implicit differentiation
- Lipschitz domain
- Rational function
- Lebesgue–Stieltjes integration
- Analytic function between Banach spaces
- Laguerre–Pólya class
- Exponential function
- Midpoint convex function
- Heaviside cover-up method
- Inversion of rational transforms by partial fractions
- Internal set theory
- Delta derivative on a time scale
- Reduction formula
- Area under a curve
- Tangent half-angle substitution
- Integration by partial fractions
- Trigonometric substitution
Theorems and results 172
- Askey–Gasper inequality
- Slopes of a convex function are increasing
- Turán's inequalities
- Bihari–LaSalle inequality
- Derivative as a little-o approximation
- Derivative of a continuous linear map
- Derivative of a map into a product
- Discrete Grönwall inequality
- Epsilon-delta definition of a limit
- Hölder's inequality for sums
- Linearity of the derivative
- Littlewood's L1 norm conjecture
- Monotone convergence theorem
- Prékopa–Leindler inequality
- Uniqueness of the derivative
- Łojasiewicz inequality
- Arithmetic–geometric mean and elliptic integrals
- Epsilon-delta definition of continuity
- Shift operator as an exponential of differentiation
- Young's inequality for integral operators
- Abel inversion formula
- Approximation property of the supremum
- Bernstein–Doetsch theorem
- Existence of square roots of nonnegative reals
- Intermediate value theorem
- Comparison of a series and an integral
- Composition of continuous functions is continuous
- De Giorgi's structure theorem
- Differentiable functions are continuous
- Dirichlet conditions
- Existence of the infimum
- Fenchel–Young inequality
- Kolmogorov–Nagumo characterization of quasi-arithmetic means
- Mean value inequality
- Remez inequality
- Sequential characterization of limits
- Young's convolution inequality
- A continuous injective function on an interval is strictly monotone
- Additivity of the integral over intervals
- Bounded derivative on a convex set implies Lipschitz
- Continuous partial derivatives imply differentiability
- Exponentials dominate powers in sequences
- Hadamard's lemma
- Linearity of the integral
- Lipschitz functions are uniformly continuous
- Monotonicity of the integral
- Product rule
- Product rule in several variables
- Reciprocal rule
- Zero derivative on a connected open set implies constant
- Chain rule in several variables
- Derivative gives the directional derivatives
- Fenchel–Moreau theorem
- Fubini's theorem
- Lusin's theorem
- Power rule
- Fundamental theorem of calculus for Lebesgue integrals
- Kneser's construction of real analytic tetration
- Linearity of differentiation
- Malgrange preparation theorem
- Sobolev inequality
- Uniformly continuous functions map Cauchy sequences to Cauchy sequences
- Carleman's condition
- Constant multiple rule
- Continuity is equivalent to sequential continuity
- Derivative vanishes at a local extremum
- Gagliardo–Nirenberg interpolation inequality
- General Leibniz rule
- One-sided limits of a monotone function exist
- C^k function characterised by continuous derivatives
- Chain rule for Jacobian matrices
- Darboux and Riemann integrals agree
- Derivative of a map into R^n by components
- Derivative of the inverse function
- Gradient points in the direction of steepest ascent
- Hardy inequality in Sobolev spaces
- Sierpiński's theorem on measurable midpoint convex functions
- A monotone function has countably many discontinuities
- Dominated convergence theorem
- Riemann's criterion for integrability
- Absolute value is multiplicative
- Changing the order of integration
- Schwarz's theorem for C^2 functions
- Triangle inequality
- C^1 functions are strictly differentiable
- Coarea formula
- Monotone functions are Riemann integrable
- Reverse triangle inequality
- Boundedness theorem
- Chain rule for a composition with a curve
- Continuity of the inverse of a strictly monotone continuous function
- Derivative of a polynomial
- Morrey's inequality
- Extreme value theorem
- Jensen inequality as a functional inequality
- Product of continuous functions is continuous
- The reals are Archimedean
- Transfer principle
- Triangle inequality for integrals
- Density of the rationals
- Mean value theorem in several variables
- Polynomial functions are continuous
- Quotient of continuous functions is continuous
- Second-order Taylor expansion
- There is a natural number n with 1/n below any positive epsilon
- Characterization of the floor
- Continuous functions on a closed bounded interval are uniformly continuous
- Continuous image of a closed interval is a closed interval
- Uniform limits commute with integration
- Gradient is orthogonal to level sets
- Lipschitz maps are uniformly continuous
- Local homeomorphism from the inverse function theorem
- Area of a region as a line integral
- Curl of a gradient is zero
- Divergence of a curl is zero
- Term-by-term integration
- Density of the irrationals
- Implicit function theorem for a surjective derivative
- Line integral as an integral over the parameter interval
- A function with nonnegative derivative is increasing
- A function with zero derivative is constant
- Additivity of line integrals under concatenation
- Reversing a path changes the sign of a line integral
- A function with bounded derivative is Lipschitz
- A function with positive derivative is strictly increasing
- Integration in polar coordinates
- A differentiable function is convex iff its derivative is increasing
- Jacobian of spherical coordinates
- Mean value theorem for divided differences
- Divergence theorem on a rectangle in the plane
- e as the limit of (1 + 1/n)^n
- Routh–Hurwitz stability criterion
- Second fundamental theorem of calculus
- Continuous functions are Riemann integrable
- First fundamental theorem of calculus
- Continuity of the sum of a uniformly convergent series of functions
- Curl-free vector field on a convex set is conservative
- Exponential growth beats polynomial growth
- Jury stability criterion
- Uniform convergence of derivatives allows term-by-term differentiation
- Jensen's inequality for integrals
- Gradient theorem
- Integral of a power
- The exponential function is its own derivative
- Vector calculus identities
- Volume of a ball in R^3
- Path independence of line integrals
- The integral of a continuous function is an antiderivative
- Hardy–Littlewood–Sobolev inequality
- Derivatives of the hyperbolic functions
- Karamata's Tauberian theorem
- Integral of the exponential function
- Power series are differentiable inside the radius of convergence
- Derivative of cosine
- Derivative of sine
- The exponential function is convex
- 1 + x is at most exp x
- The logarithm grows slower than any power
- Derivative of the logarithm
- Derivative of x^a
- Integral of cosine
- Integral of sine
- Homotopy invariance of line integrals of closed forms
- The logarithm is strictly concave
- Derivative of tangent
- Integral of 1/x
- Poincaré inequality
- log x is at most x - 1
- Derivative of arcsine
- Harmonic numbers grow like log n
- Derivative of arctangent
- Log-convexity of the gamma function
Examples 7
Counterexamples 12
- sin(1/x) has no limit at 0
- x^2 is not uniformly continuous on the real line
- Dirichlet function
- Absolute value is continuous but not differentiable at 0
- Function for which the iterated integrals differ
- Separately continuous but not continuous function
- Partial derivatives do not imply differentiability
- Function with unequal mixed partial derivatives
- The Dirichlet function is not Riemann integrable
- x^2 sin(1/x) is differentiable with discontinuous derivative
- A continuous nowhere differentiable function
- Curl-free field that is not conservative
Objects 3
Problems 1
Definitions 200
- Convex function
- Even and odd functions
- Limit point of a set
- Lipschitz continuity
- Local maximum
- Luzin N property
- Partition of an interval
- Periodic function
- Strictly monotone function
- Arc length
- Cauchy principal value
- Concave function
- Having a directional derivative
- Limit of a function
- Lipschitz condition
- Littlewood polynomial
- Local minimum
- Majorization
- Mesh of a partition
- Refinement of a partition
- Riemann sum
- Riemann–Liouville integral
- Schur stable polynomial
- Slowly varying function
- Step function
- Strictly convex function
- Abel equation
- Abel function
- Bounded variation
- Cobweb plot
- Completely monotone function
- Continuity at a point
- Derivative
- Epigraph
- Functional square root
- Grünwald–Letnikov derivative
- Hadamard finite part
- Henstock–Kurzweil integral
- Homogeneity equation
- Infinite limit
- Inflection point
- Iterated function
- Iterative root
- Jensen's functional equation
- Lebesgue integral
- Limit at infinity
- One-sided limit
- Regular variation
- Schur-convex function
- Stieltjes function
- Absolutely monotonic function
- Antiderivative
- Approximate derivative
- Caputo fractional derivative
- Continuous function
- Convex function on R^n
- Differentiable function
- Fractional calculus
- Integrable function
- Iterated integral
- Iterative functional equation
- Jump discontinuity
- Lipschitz class
- Local maximum and minimum
- Loeb measure
- Mittag-Leffler function
- Modulus of smoothness
- Removable discontinuity
- Riemann integrable function
- Riesz potential
- Sequential continuity
- Space of functions of bounded variation
- Subadditive function
- Superadditive function
- Time-scale calculus
- Total variation
- Clarke generalized gradient
- Continuity in R^n
- Derivative function
- Differentiable function of several variables
- Directional derivative
- Fenchel conjugate
- Indefinite integral
- Lebesgue–Stieltjes measure
- Level set
- Limit of a function of several variables
- One-sided derivative
- P-adic analysis
- Partial derivative
- Quasi-arithmetic mean
- Riemann integral
- Sine and cosine
- Standard part function
- Uniform continuity
- Upper and lower Darboux sums
- Convex integrand
- Definite integral
- Differentiable on an open set
- Double integral
- Fractional differential equation
- Having a gradient
- Hemicontinuity
- Higher derivative
- Higher partial derivatives
- Lagrange polynomial
- Monad in nonstandard topology
- Rank-one convexity
- Riemann–Stieltjes integral
- Semicontinuity
- Singular function
- Square root
- Uniform continuity on a set
- Upper and lower Darboux integrals
- Volterra equations with completely monotone kernels
- Darboux integral
- Dedekind cut
- Germ of a smooth function
- Gradient
- Hessian matrix
- Jacobian matrix
- Khinchin integral
- Morrey quasiconvexity
- Perron integral
- Second derivative
- Square root function
- Strict differentiability
- Symmetric decreasing rearrangement
- Trigonometric functions
- Triple integral
- Caccioppoli set
- Centre of mass
- Dynamic equation on time scales
- Fractional iteration
- Hyperreal number
- Jacobian determinant
- Littlewood's three principles
- Moment of inertia
- Order on the reals
- Polyconvexity
- Tangent plane to a graph
- Continuously differentiable function
- Fractional integration
- Interval
- Multiple integral
- Polynomial function
- Quasiconvex function
- Saddle point
- Absolute continuity
- Improper integral
- Iterated derivative
- Smooth function
- Laplace–Stieltjes transform
- Perimeter of a set
- Absolute value
- Dini derivative
- Fractional Laplacian
- Subderivative
- Vector field
- Curl
- Laplace's method
- Rectifiable curve
- Line integral of a scalar field
- Lipschitz map
- Conservative vector field
- Divergence
- Floor function
- Hölder continuity
- Legendre transformation
- Line integral of a vector field
- Ceiling function
- Scalar potential
- Symmetric derivative
- Nonstandard analysis
- Lipschitz domain
- Rational function
- Lebesgue–Stieltjes integration
- Quasi-analytic function
- Analytic function between Banach spaces
- Laguerre–Pólya class
- Surface integral
- Exponential function
- Midpoint convex function
- Flux through a surface
- Heaviside cover-up method
- Inversion of rational transforms by partial fractions
- Modulus of continuity
- Smoothness classes
- Internal set theory
- Oscillation of a function
- Delta derivative on a time scale
- Operator monotone function
- Semi-continuity
- Difference quotient
- Maximum and minimum
- Bump function
- Darboux function
- Natural logarithm
- Real exponentiation
- Baire function
- Logarithmically convex function