List of real analysis topics

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

General topics[edit]

Limits[edit]

Sequences and series[edit]

(see also list of mathematical series)

Summation methods[edit]

More advanced topics[edit]

  • 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.

Convergence[edit]

Convergence tests[edit]

Functions[edit]

Continuity[edit]

Distributions[edit]

Variation[edit]

Derivatives[edit]

Differentiation rules[edit]

Differentiation in geometry and topology[edit]

see also List of differential geometry topics

Integrals[edit]

(see also Lists of 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

Integration and measure theory[edit]

see also List of integration and measure theory topics

Fundamental theorems[edit]

  • 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.

Foundational topics[edit]

Numbers[edit]

Real numbers[edit]

Specific numbers[edit]

Sets[edit]

Maps[edit]

Applied mathematical tools[edit]

Infinite expressions[edit]

Inequalities[edit]

See list of inequalities

Means[edit]

Orthogonal polynomials[edit]

Spaces[edit]

Measures[edit]

  • 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[edit]

Historical figures[edit]

Related fields of analysis[edit]