Editing List of real analysis topics

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

See also: [[glossary of real and complex analysis]].

==General topics==

===[[Limit (mathematics)|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

===[[Sequence]]s and [[Series (mathematics)|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 (mathematics)|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 <math>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</math>
**[[Taylor series]] – a series of the form <math>f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots. </math>
***'''Maclaurin series''' – ''see [[Taylor series]]''
****[[Binomial series]] – the Maclaurin series of the function ''f'' given by ''f''(''x'')&nbsp;''=''&nbsp;(1&nbsp;+&nbsp;''x'')<sup>&nbsp;''&alpha;''</sup>
*[[Telescoping series]]
*[[Alternating series]]
*[[Geometric series]]
**[[Divergent geometric series]]
*[[Harmonic series (mathematics)|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 fraction]]s between 0 and 1
*[[Oscillation (mathematics)|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 form]]s – algebraic expressions gained in the context of limits. The indeterminate forms include 0<sup>0</sup>, 0/0, 1<sup>∞</sup>, ∞&nbsp;&minus;&nbsp;∞, ∞/∞,  0&nbsp;×&nbsp;∞, and ∞<sup>0</sup>.

===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

===[[Function (mathematics)|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 of a function|Domain]]
*[[Codomain]]
**[[Image (mathematics)|Image]]
*[[Support (mathematics)|Support]]
*[[Differential of a function]]

====Continuity====

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

====[[distribution (mathematics)|Distribution]]s====

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

====Variation====

*[[Bounded variation]]
*[[Total variation]]

===[[Derivative]]s===

*[[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 (mathematics)|Submersion]] – a differentiable map between differentiable manifolds whose differential is everywhere surjective

===[[Integral]]s===

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

==Foundational topics==

===[[Number]]s===

====[[Real number]]s====

*[[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 (number)|0]]
*[[1 (number)|1]]
**[[0.999...]]
*[[Infinity]]

===[[Set (mathematics)|Sets]]===

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

===[[Map (mathematics)|Maps]]===

*[[Contraction mapping]]
*[[Metric map]]
*[[Fixed point (mathematics)|Fixed point]] – a point of a function that maps to itself

==Applied mathematical tools==

===[[Infinite expression (mathematics)|Infinite expressions]]===

*[[Continued fraction]]
*[[Series (mathematics)|Series]]
*[[Infinite product]]s

===[[Inequality (mathematics)|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]]

===[[Mean]]s===
*[[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]]

===[[Space (mathematics)|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]]

===[[Measure (mathematics)|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)

==[[Mathematical analysis|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]].

==See also==
* [[Calculus]], the classical calculus of [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]].
* [[Non-standard calculus]], a rigorous application of [[infinitesimals]], in the sense of [[non-standard analysis]], to the classical calculus of Newton and Leibniz.

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