List of real analysis topics

This is a list of articles that are considered real analysis topics.

See also: glossary of real and complex analysis.

## General topics

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### Limits

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* Limit of a sequence
* Subsequential limit – the limit of some subsequence
* Limit of a function (see List of limits for a list of limits of common functions)
* One-sided limit – either of the two limits of functions of real variables x, as x approaches a point from above or below
* Squeeze theorem – confirms the limit of a function via comparison with two other functions
* Big O notation – used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions

### Sequences and series

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(see also list of mathematical series)

* Arithmetic progression – a sequence of numbers such that the difference between the consecutive terms is constant
* Generalized arithmetic progression – a sequence of numbers such that the difference between consecutive terms can be one of several possible constants
* Geometric progression – a sequence of numbers such that each consecutive term is found by multiplying the previous one by a fixed non-zero number
* Harmonic progression – a sequence formed by taking the reciprocals of the terms of an arithmetic progression
* Finite sequence – see sequence
* Infinite sequence – see sequence
* Divergent sequence – see limit of a sequence or divergent series
* Convergent sequence – see limit of a sequence or convergent series
* Cauchy sequence – a sequence whose elements become arbitrarily close to each other as the sequence progresses
* Convergent series – a series whose sequence of partial sums converges
* Divergent series – a series whose sequence of partial sums diverges
* Power series – a series of the form  f ( x ) = ∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) 1 + a 2 ( x − c ) 2 + a 3 ( x − c ) 3 + ⋯ {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots }
* Taylor series – a series of the form  f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ( 3 ) ( a ) 3 ! ( x − a ) 3 + ⋯ . {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots .}
* Maclaurin series – see Taylor series
* Binomial series – the Maclaurin series of the function f given by f(x) = (1 + x) α
* Telescoping series
* Alternating series
* Geometric series
* Divergent geometric series
* Harmonic series
* Fourier series
* Lambert series

#### Summation methods

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* Cesàro summation
* Euler summation
* Lambert summation
* Borel summation
* Summation by parts – transforms the summation of products of into other summations
* Cesàro mean
* Abel's summation formula

#### More advanced topics

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* Convolution
* Cauchy product –is the discrete convolution of two sequences
* Farey sequence – the sequence of completely reduced fractions between 0 and 1
* Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.
* Indeterminate forms – algebraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1∞, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0.

### Convergence

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* Pointwise convergence, Uniform convergence
* Absolute convergence, Conditional convergence
* Normal convergence
* Radius of convergence

#### Convergence tests

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* Integral test for convergence
* Cauchy's convergence test
* Ratio test
* Direct comparison test
* Limit comparison test
* Root test
* Alternating series test
* Dirichlet's test
* Stolz–Cesàro theorem – is a criterion for proving the convergence of a sequence

### Functions

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* Function of a real variable
* Real multivariable function
* Continuous function
* Nowhere continuous function
* Weierstrass function
* Smooth function
* Analytic function
* Quasi-analytic function
* Non-analytic smooth function
* Flat function
* Bump function
* Differentiable function
* Integrable function
* Square-integrable function, p-integrable function
* Monotonic function
* Bernstein's theorem on monotone functions – states that any real-valued function on the half-line [0, ∞) that is totally monotone is a mixture of exponential functions
* Inverse function
* Convex function, Concave function
* Singular function
* Harmonic function
* Weakly harmonic function
* Proper convex function
* Rational function
* Orthogonal function
* Implicit and explicit functions
* Implicit function theorem – allows relations to be converted to functions
* Measurable function
* Baire one star function
* Symmetric function
* Domain
* Codomain
* Image
* Support
* Differential of a function

#### Continuity

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* Uniform continuity
* Modulus of continuity
* Lipschitz continuity
* Semi-continuity
* Equicontinuous
* Absolute continuity
* Hölder condition – condition for Hölder continuity

#### Distributions

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* Dirac delta function
* Heaviside step function
* Hilbert transform
* Green's function

#### Variation

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* Bounded variation
* Total variation

### Derivatives

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* Second derivative
* Inflection point – found using second derivatives
* Directional derivative, Total derivative, Partial derivative

#### Differentiation rules

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* Linearity of differentiation
* Product rule
* Quotient rule
* Chain rule
* Inverse function theorem – gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain, also gives a formula for the derivative of the inverse function

#### Differentiation in geometry and topology

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see also List of differential geometry topics

* Differentiable manifold
* Differentiable structure
* Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

### Integrals

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(see also Lists of integrals)

* Antiderivative
* Fundamental theorem of calculus – a theorem of antiderivatives
* Multiple integral
* Iterated integral
* Improper integral
* Cauchy principal value – method for assigning values to certain improper integrals
* Line integral
* Anderson's theorem – says that the integral of an integrable, symmetric, unimodal, non-negative function over an n-dimensional convex body (K) does not decrease if K is translated inwards towards the origin

#### Integration and measure theory

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see also List of integration and measure theory topics

* Riemann integral, Riemann sum
* Riemann–Stieltjes integral
* Darboux integral
* Lebesgue integration

## Fundamental theorems

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* Monotone convergence theorem – relates monotonicity with convergence
* Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
* Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
* Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
* Taylor's theorem – gives an approximation of a  k {\displaystyle k} times differentiable function around a given point by a  k {\displaystyle k} -th order Taylor-polynomial.
* L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
* Abel's theorem – relates the limit of a power series to the sum of its coefficients
* Lagrange inversion theorem – gives the Taylor series of the inverse of an analytic function
* Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
* Heine–Borel theorem – sometimes used as the defining property of compactness
* Bolzano–Weierstrass theorem – states that each bounded sequence in  R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence
* Extreme value theorem \- states that if a function  f {\displaystyle f} is continuous in the closed and bounded interval  [ a , b ] {\displaystyle [a,b]} , then it must attain a maximum and a minimum

## Foundational topics

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### Numbers

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#### Real numbers

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* Construction of the real numbers
* Natural number
* Integer
* Rational number
* Irrational number
* Completeness of the real numbers
* Least-upper-bound property
* Real line
* Extended real number line
* Dedekind cut

#### Specific numbers

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* 0
* 1
* 0.999...
* Infinity

### Sets

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* Open set
* Neighbourhood
* Cantor set
* Derived set (mathematics)
* Completeness
* Limit superior and limit inferior
* Supremum
* Infimum
* Interval
* Partition of an interval

### Maps

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* Contraction mapping
* Metric map
* Fixed point – a point of a function that maps to itself

## Applied mathematical tools

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### Infinite expressions

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* Continued fraction
* Series
* Infinite products

### Inequalities

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See list of inequalities

* Triangle inequality
* Bernoulli's inequality
* Cauchy–Schwarz inequality
* Hölder's inequality
* Minkowski inequality
* Jensen's inequality
* Chebyshev's inequality
* Inequality of arithmetic and geometric means

### Means

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* Generalized mean
* Pythagorean means
* Arithmetic mean
* Geometric mean
* Harmonic mean
* Geometric–harmonic mean
* Arithmetic–geometric mean
* Weighted mean
* Quasi-arithmetic mean

### Orthogonal polynomials

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* Classical orthogonal polynomials
* Hermite polynomials
* Laguerre polynomials
* Jacobi polynomials
* Gegenbauer polynomials
* Legendre polynomials

### Spaces

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* Euclidean space
* Metric space
* Banach fixed point theorem – guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, provides method to find them
* Complete metric space
* Topological space
* Function space
* Sequence space
* Compact space

### Measures

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* Lebesgue measure
* Outer measure
* Hausdorff measure
* Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

### Field of sets

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* Sigma-algebra

## Historical figures

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* Michel Rolle (1652–1719)
* Brook Taylor (1685–1731)
* Leonhard Euler (1707–1783)
* Joseph-Louis Lagrange (1736–1813)
* Joseph Fourier (1768–1830)
* Bernard Bolzano (1781–1848)
* Augustin Cauchy (1789–1857)
* Niels Henrik Abel (1802–1829)
* Peter Gustav Lejeune Dirichlet (1805–1859)
* Karl Weierstrass (1815–1897)
* Eduard Heine (1821–1881)
* Pafnuty Chebyshev (1821–1894)
* Leopold Kronecker (1823–1891)
* Bernhard Riemann (1826–1866)
* Richard Dedekind (1831–1916)
* Rudolf Lipschitz (1832–1903)
* Camille Jordan (1838–1922)
* Jean Gaston Darboux (1842–1917)
* Georg Cantor (1845–1918)
* Ernesto Cesàro (1859–1906)
* Otto Hölder (1859–1937)
* Hermann Minkowski (1864–1909)
* Alfred Tauber (1866–1942)
* Felix Hausdorff (1868–1942)
* Émile Borel (1871–1956)
* Henri Lebesgue (1875–1941)
* Wacław Sierpiński (1882–1969)
* Johann Radon (1887–1956)
* Karl Menger (1902–1985)

## Related fields of analysis

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* Asymptotic analysis – studies a method of describing limiting behaviour
* Convex analysis – studies the properties of convex functions and convex sets
* List of convexity topics
* Harmonic analysis – studies the representation of functions or signals as superpositions of basic waves
* List of harmonic analysis topics
* Fourier analysis – studies Fourier series and Fourier transforms
* List of Fourier analysis topics
* List of Fourier-related transforms
* Complex analysis – studies the extension of real analysis to include complex numbers
* Functional analysis – studies vector spaces endowed with limit-related structures and the linear operators acting upon these spaces
* Nonstandard analysis – studies mathematical analysis using a rigorous treatment of infinitesimals.

## See also

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* Calculus, the classical calculus of Newton and Leibniz.
* Non-standard calculus, a rigorous application of infinitesimals, in the sense of non-standard analysis, to the classical calculus of Newton and Leibniz.