List of real analysis topics

General topics

Limits">Limit (mathematics)">Limits

• Limit of a sequence
• *Subsequential limit – the limit of some subsequence
• Limit of a function
• *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">Series (mathematics)">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 = ^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 0⁰, 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 (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
• 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

  Distribution">distribution (mathematics)">Distributions

• 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

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

  [Integral]s

• 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 does not decrease if K is translated inwards towards the origin

  Integration and measure theory

• 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 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

[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
• 1
• *0.999...
• Infinity

  Sets">Set (mathematics)">Sets

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

  Maps">Map (mathematics)">Maps

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

  Applied mathematical tools

Infinite expressions">Infinite expression (mathematics)">Infinite expressions

• Continued fraction
• Series
• Infinite products

  Inequalities">Inequality (mathematics)">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

  Spaces">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

  Measures">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
• Brook Taylor
• Leonhard Euler
• Joseph-Louis Lagrange
• Joseph Fourier
• Bernard Bolzano
• Augustin Cauchy
• Niels Henrik Abel
• Peter Gustav Lejeune Dirichlet
• Karl Weierstrass
• Eduard Heine
• Pafnuty Chebyshev
• Leopold Kronecker
• Bernhard Riemann
• Richard Dedekind
• Rudolf Lipschitz
• Camille Jordan
• Jean Gaston Darboux
• Georg Cantor
• Ernesto Cesàro
• Otto Hölder
• Hermann Minkowski
• Alfred Tauber
• Felix Hausdorff
• Émile Borel
• Henri Lebesgue
• Wacław Sierpiński
• Johann Radon
• Karl Menger

  Related fields of analysis">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.