# 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
### Limits

* 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

(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 $f(x)=\sum _{n=0}^{\infty }a_{n}\left(x-c\right)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+a_{3}(x-c)^{3}+\cdots$
** Taylor series – a series of the form $f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f^{(3)}(a)}{3!}}(x-a)^{3}+\cdots .$
*** 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

* 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

* 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 0$^0$, 0/0, 1$^∞$, ∞ − ∞, ∞/∞, 0 × ∞, and ∞$^0$.

### Convergence

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

#### Convergence tests

* 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

* 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

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

#### Distributions

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

#### Variation

* Bounded variation
* Total variation

### Derivatives

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

#### Differentiation rules

* 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

see also List of differential geometry topics
* Differentiable manifold
* Differentiable structure
* Submersion – a differentiable map between differentiable manifolds whose differential is everywhere surjective

### Integrals

(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

see also List of integration and measure theory topics
* Riemann integral, Riemann sum
** Riemann–Stieltjes integral
* Darboux integral
* Lebesgue integration

## Fundamental theorems

* 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 $k$ times differentiable function around a given point by a $k$-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 $\mathbb {R} ^{n}$ has a convergent subsequence
* Extreme value theorem - states that if a function $f$ is continuous in the closed and bounded interval $[a,b]$, then it must attain a maximum and a minimum

## Foundational topics
### Numbers
#### Real numbers

* 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

* 0
* 1
** 0.999...
* Infinity

### Sets

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

### Maps

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

## Applied mathematical tools
### Infinite expressions

* Continued fraction
* Series
* Infinite products

### Inequalities

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

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

### Orthogonal polynomials

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

### Spaces

* 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

* 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

* Sigma-algebra

## Historical figures

* 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

* 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.