List of real analysis topics

This is a list of articles that are considered real analysis 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)&alpha;
 * 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 00, 0/0, 1∞, ∞ &minus; ∞, ∞/∞, 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

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

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.