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

## 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 $\mathbb {R} ^{n}$  has a convergent subsequence
• Extreme value theorem - states that if a function $f$  is continuous in the closed and bounded interval $[a,b]$ , then it must attain a maximum and a minimum