Real analysis

value that a function (or sequence) approaches as the argument (or index) approaches some value

expression of a function as an infinite sum

upright=1.35|class=skin-invert-image|thumb|alt=Stylistic impression of the repeating decimal 0.9999..., representing the digit 9 repeating infinitely 0.999... is a repeating decimal that represents the number 1. The three dots represent an infinite list of "9" digits. Following the standard rules for representing real numbers in decimal notation, its value is the smallest number greater than or equal to every number in the sequence 0.9, 0.99, 0.999, and so on. It can be proved that this number is1; that is, 0.999\ldots = 1.

branch of mathematical analysis

infinite sum of monomials

function between ordered sets that preserves or reverses the given order

function that returns 1 if an element is present in a specified subset and 0 if absent; naturally isomorphic with a set's subsets

infinite series in which the signs of the general terms alternate between positive and negative

indicator function of rational numbers

function that is continuous everywhere but differentiable nowhere

function or sequence whose possible values form a bounded set

part of the domain of a mathematical function

every element of a partially ordered set A which is greater (resp. lower) than every element of a subset B included in A

Taylor series

concept in mathematics

form of continuity for functions

peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity

inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions

In mathematics, a càdlàg (), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset of them) that is everywhere right-continuous and has left limits everywhere. Càdlàg functions are important in the study of stochastic processes that admit (or even require) jumps, unlike Brownian motion, which has continuous sample paths. The collection of càdlàg functions on a given domain is known as Skorokhod space.

limit of a function approaching a value point from values below or above the value point

theorem to simplify sums of products of sequences

real-valued function composed of straight line segments

piecewise function that clamps its input to be non-negative

complex function on a measurable space that is piecewise constant with a finite number of measurable regions

theorem that a decreasing nested sequences of nonempty closed compact sets has nonempty intersection

function that "converges" to periodicity

real function with finite total variation

function thats composition with the logarithm is a convex function

function for which every set of inputs whose value is below a given threshold is convex

functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions

theorem

inequality

class of generalisations of the derivative

mathematical property shared by many number systems, that a product cannot be zero unless one of its factors is zero

topological space

property of a partially ordered set

equation in mathematics

integral operator whose kernel is singular along the diagonal

Mathematical operator in real and harmonic analysis

shape containing unit line segments in all directions

first article on transfinite set theory

Measure theory concept

Type of mathematical function

extension of the set of the real numbers by a point denoted ∞

function in mathematics

heuristics in measure theory

amount of variation between extrema of a function or sequence