List of real analysis topics

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

General topics

Limits

Sequences and series

(see also list of mathematical series)

Summation methods

More advanced topics

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

Convergence

Convergence tests

Functions

Continuity

Distributions

Variation

Derivatives

Differentiation rules

Differentiation in geometry and topology

see also List of differential geometry topics

Integrals

(see also Lists of integrals)

Integration and measure theory

see also List of integration and measure theory topics

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

Real numbers

Specific numbers

Sets

Maps

Applied mathematical tools

Infinite expressions

Inequalities

See list of inequalities

Means

Orthogonal polynomials

Spaces

Measures

Field of sets

Historical figures

See also

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