Editing Real analysis (section)

===Continuity===
{{Main|Continuous function}}
A [[function (mathematics)|function]] from the set of [[real number]]s to the real numbers can be represented by a [[graph of a function|graph]] in the [[Cartesian coordinate system|Cartesian plane]]; such a function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] with no "holes" or "jumps".

There are several ways to make this intuition mathematically rigorous. Several definitions of varying levels of generality can be given.  In cases where two or more definitions are applicable, they are readily shown to be [[Equivalence relation|equivalent]] to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. In the first definition given below, <math>f:I\to\R</math> is a function defined on a non-degenerate interval <math>I</math> of the set of real numbers as its domain. Some possibilities include <math>I=\R</math>, the whole set of real numbers, an [[open interval]] <math>I = (a, b) = \{x \in \R \mid a < x < b \}, </math> or a [[closed interval]] <math>I = [a, b] = \{x \in \R \mid a \leq x \leq b\}. </math>  Here, <math>a</math> and <math>b</math> are distinct real numbers, and we exclude the case of <math>I</math> being empty or consisting of only one point, in particular.

'''Definition.'''  If <math>I\subset \mathbb{R}</math> is a non-degenerate interval, we say that <math>f:I \to \R</math> is '''''continuous at''''' <math>p\in I</math> if <math display="inline">\lim_{x \to p} f(x) = f(p)</math>.  We say that <math>f</math> is a '''''continuous map''''' if <math>f</math> is continuous at every <math>p\in I</math>.

In contrast to the requirements for <math>f</math> to have a limit at a point <math>p</math>, which do not constrain the behavior of <math>f</math> at <math>p</math> itself, the following two conditions, in addition to the existence of <math display="inline">\lim_{x\to p} f(x)</math>, must also hold in order for <math>f</math> to be continuous at <math>p</math>: '''(i)''' <math>f</math> must be defined at <math>p</math>, i.e., <math>p</math> is in the domain of <math>f</math>; ''and'' '''(ii)''' <math>f(x)\to f(p)</math> as <math>x\to p</math>.  The definition above actually applies to any domain <math>E</math> that does not contain an [[isolated point]], or equivalently, <math>E</math> where every <math>p\in E</math> is a [[limit point]] of <math>E</math>.  A more general definition applying to <math>f:X\to\mathbb{R}</math> with a general domain <math>X\subset \mathbb{R}</math> is the following:

'''Definition.'''  If <math>X</math> is an arbitrary subset of <math>\mathbb{R}</math>, we say that <math>f:X\to\mathbb{R}</math> is '''''continuous at''''' <math>p\in X</math> if, for any <math>\varepsilon>0</math>, there exists <math>\delta>0</math> such that for all <math>x\in X</math>, <math>|x-p|<\delta</math> implies that <math>|f(x)-f(p)| < \varepsilon</math>.  We say that <math>f</math> is a '''''continuous map''''' if <math>f</math> is continuous at every <math>p\in X</math>.

A consequence of this definition is that <math>f</math> is ''trivially continuous at any isolated point'' <math>p\in X</math>.  This somewhat unintuitive treatment of isolated points is necessary to ensure that our definition of continuity for functions on the real line is consistent with the most general definition of continuity for maps between [[topological space]]s (which includes [[metric space]]s and <math>\mathbb{R}</math> in particular as special cases).  This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.

'''Definition.'''  If <math>X</math> and <math>Y</math> are topological spaces, we say that <math>f:X\to Y</math> is '''''continuous at''''' <math>p\in X</math> if <math>f^{-1} (V)</math> is a [[Neighborhood (topology)|neighborhood]] of <math>p</math> in <math>X</math> for every neighborhood <math>V</math> of <math>f(p)</math> in <math>Y</math>.  We say that <math>f</math> is a '''''continuous map''''' if <math>f^{-1}(U)</math> is open in <math>X</math> for every <math>U</math> open in <math>Y</math>.

(Here, <math>f^{-1}(S)</math> refers to the [[preimage]] of <math>S\subset Y</math> under <math>f</math>.)

====Uniform continuity====
{{Main|Uniform continuity}}
'''Definition.''' If <math>X</math> is a subset of the [[real number]]s, we say a function <math>f:X\to\mathbb{R}</math> is '''''uniformly continuous''''' '''''on''''' <math>X</math> if, for any <math>\varepsilon > 0</math>, there exists a <math>\delta>0</math> such that for all <math>x,y\in X</math>, <math>|x-y|<\delta</math> implies that <math>|f(x)-f(y)| < \varepsilon</math>.

Explicitly, when a function is uniformly continuous on <math>X</math>, the choice of <math>\delta</math> needed to fulfill the definition must work for ''all of'' <math>X</math> for a given <math>\varepsilon</math>.  In contrast, when a function is continuous at every point <math>p\in X</math> (or said to be continuous on <math>X</math>), the choice of <math>\delta</math> may depend on both <math>\varepsilon</math> ''and'' <math>p</math>.  In contrast to simple continuity, uniform continuity is a property of a function that only makes sense with a specified domain; to speak of uniform continuity at a single point <math>p</math> is meaningless.

On a compact set, it is easily shown that all continuous functions are uniformly continuous.  If <math>E</math> is a bounded noncompact subset of <math>\mathbb{R}</math>, then there exists <math>f:E\to\mathbb{R}</math> that is continuous but not uniformly continuous.  As a simple example, consider <math>f:(0,1)\to\mathbb{R}</math> defined by <math>f(x)=1/x</math>.  By choosing points close to 0, we can always make <math>|f(x)-f(y)| > \varepsilon</math> for any single choice of <math>\delta>0</math>, for a given <math>\varepsilon > 0</math>.

====Absolute continuity====
{{Main|Absolute continuity}}
'''Definition.''' Let <math>I\subset\mathbb{R}</math> be an [[interval (mathematics)|interval]] on the [[real line]]. A function <math>f:I \to \mathbb{R}</math> is said to be '''''absolutely continuous''''' '''''on''''' <math>I</math> if for every positive number <math>\varepsilon</math>, there is a positive number <math>\delta</math> such that whenever a finite sequence of [[pairwise disjoint]] sub-intervals <math>(x_1, y_1), (x_2,y_2),\ldots, (x_n,y_n)</math> of <math>I</math> satisfies<ref>{{harvnb|Royden|1988|loc=Sect. 5.4, page 108}}; {{harvnb|Nielsen|1997|loc=Definition 15.6 on page 251}}; {{harvnb|Athreya|Lahiri|2006|loc=Definitions 4.4.1, 4.4.2 on pages 128,129}}. The interval ''I'' is assumed to be bounded and closed in the former two books but not the latter book.</ref>
:<math>\sum_{k=1}^{n} (y_k - x_k) < \delta</math>
then
:<math>\sum_{k=1}^{n} | f(y_k) - f(x_k) | < \varepsilon.</math>
Absolutely continuous functions are continuous: consider the case ''n'' = 1 in this definition.  The collection of all absolutely continuous functions on ''I'' is denoted AC(''I'').  Absolute continuity is a fundamental concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral.