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

## 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
- Taylor series – a series of the form
**Maclaurin series**–*see Taylor series*- Binomial series – the Maclaurin series of the function
*f*given by*f*(*x*)*=*(1 +*x*)^{ α}

- Binomial series – the Maclaurin series of the function

- Taylor series – a series of the form
- Telescoping series
- Alternating series
- 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
- Smooth function
- Differentiable function
- 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
- 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
- Support
- Differential of a function

#### Continuity

- Uniform continuity
- Lipschitz continuity
- Semi-continuity
- Equicontinuous
- Absolute continuity
- Hölder condition – condition for Hölder continuity

#### Distributions

#### 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*

## 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 times differentiable function around a given point by a -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 has a convergent subsequence**Extreme value theorem**- states that if a function is continuous in the closed and bounded interval , then it must attain a maximum and a minimum

## Foundational topics

### Numbers

#### Real numbers

- Construction of the real numbers
- Completeness of the real numbers
- Least-upper-bound property
- Real line

#### Specific numbers

### Sets

- Open set
- Neighbourhood
- Cantor set
- Derived set (mathematics)
- Completeness
- Limit superior and limit inferior
- Interval

### Maps

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

## Applied mathematical tools

### Infinite expressions

### 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
- Geometric–harmonic mean
- Arithmetic–geometric mean
- Weighted mean
- Quasi-arithmetic mean

### Orthogonal 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
- Compact space

### Measures

- Lebesgue measure
- Outer 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

## 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**Harmonic analysis**– studies the representation of functions or signals as superpositions of basic waves**Fourier analysis**– studies Fourier series and Fourier 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

- 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.