List of real analysis topics

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

General topics

Sequences and series

Summation methods

• Convolution
• 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 – algerbraic expressions gained in the context of limits. The indeterminate forms include 00, 0/0, 1, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0.

Integrals

• 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

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 Rn has a convergent subsequence.

Applied mathematical tools

Inequalities

See list of inequalities

Measures

• Dominated convergence theorem – provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions.

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