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 ) = ∑ n = 0 ∞ a n ( x − c ) n = a 0 + a 1 ( x − c ) 1 + a 2 ( x − c ) 2 + a 3 ( x − c ) 3 + ⋯ {\displaystyle 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 ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ( 3 ) ( a ) 3 ! ( x − a ) 3 + ⋯ . {\displaystyle 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 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 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 {\displaystyle k} times differentiable function around a given point by a k {\displaystyle 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 R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence Extreme value theorem - states that if a function f {\displaystyle f} is continuous in the closed and bounded interval [ a , b ] {\displaystyle [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.
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.