List of real analysis topics

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

General topicsEdit


Sequences and seriesEdit

(see also list of mathematical series)

Summation methodsEdit

More advanced topicsEdit

  • 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 testsEdit






Differentiation rulesEdit

Differentiation in geometry and topologyEdit

see also List of differential geometry topics


(see also Lists of integrals)

Integration and measure theoryEdit

see also List of integration and measure theory topics

Fundamental theoremsEdit

  • 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 topicsEdit


Real numbersEdit

Specific numbersEdit



Applied mathematical toolsEdit

Infinite expressionsEdit


See list of inequalities


Orthogonal polynomialsEdit



Field of setsEdit

Historical figuresEdit

Related fields of analysisEdit

See alsoEdit