Editing Continuous function (section)

===Definition===
[[File:Function-1 x.svg|thumb|The function <math>f(x)=\tfrac 1 x</math> is continuous on its domain (<math>\R\setminus \{0\}</math>), but is discontinuous at <math>x=0,</math> when considered as a [[partial function]] defined on the reals.<ref>{{cite book |last1=Strang |first1=Gilbert |title=Calculus |year=1991 |publisher=SIAM|isbn=0961408820 |page=702|url={{Google books|OisInC1zvEMC|page=87|plainurl=yes}}}}</ref>]]

A [[real function]] that is a [[function (mathematics)|function]] from [[real number]]s to 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]] whose [[domain of a function|domain]] is the entire real line. A more mathematically rigorous definition is given below.<ref>{{cite web | url=http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf | title=Continuity and Discontinuity | last1=Speck | first1=Jared | year=2014 | page=3 | access-date=2016-09-02 | website=MIT Math | quote=Example 5. The function <math>1/x</math> is continuous on <math>(0, \infty)</math> and on <math>(-\infty, 0),</math>, i.e., for <math>x > 0</math> and for <math>x < 0,</math> in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely <math>x = 0,</math>, and an infinite discontinuity there. | archive-date=2016-10-06 | archive-url=https://web.archive.org/web/20161006014646/http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf | url-status=dead }}</ref>

Continuity of real functions is usually defined in terms of [[Limit (mathematics)|limits]]. A function {{math|''f''}} with variable {{mvar|x}} is ''continuous at'' the [[real number]] {{mvar|c}}, if the limit of <math>f(x),</math> as {{mvar|x}} tends to {{mvar|c}}, is equal to <math>f(c).</math>

There are several different definitions of the (global) continuity of a function, which depend on the nature of its [[domain of a function|domain]]. 

A function is continuous on an [[open interval]] if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval <math>(-\infty, +\infty)</math> (the whole [[real line]]) is often called simply a continuous function; one also says that such a function is ''continuous everywhere''. For example, all [[polynomial function]]s are continuous everywhere.

A function is continuous on a [[semi-open interval|semi-open]] or a [[closed interval|closed]] interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function <math>f(x) = \sqrt{x}</math> is continuous on its whole domain, which is the closed interval <math>[0,+\infty).</math>

Many commonly encountered functions are [[partial function]]s that have a domain formed by all real numbers, except some [[isolated point]]s. Examples include the [[reciprocal function]] <math display="inline">x \mapsto \frac {1}{x}</math> and the [[tangent function]] <math>x\mapsto \tan x.</math> When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.

A partial function is ''discontinuous'' at a point if the point belongs to the [[topological closure]] of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions <math display="inline">x\mapsto \frac {1}{x}</math> and <math display="inline">x\mapsto \sin(\frac {1}{x})</math> are discontinuous at {{math|0}}, and remain discontinuous whichever value is chosen for defining them at {{math|0}}. A point where a function is discontinuous is called a ''discontinuity''.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Let <math display=inline>f : D \to \R</math> be a function whose [[domain of a function|domain]]  <math>D</math> is contained in <math>\R</math> of real numbers.

Some (but not all) possibilities for <math>D</math> are: 
*<math> D </math> is the whole [[real line]]; that is, <math>D = \R </math>
*<math> D </math> is a [[closed interval]] of the form <math>D = [a, b] = \{x \in \R \mid a \leq x \leq b \} ,</math> where {{mvar|a}} and {{mvar|b}} are real numbers
*<math> D </math> is an [[open interval]] of the form <math>D = (a, b) = \{x \in \R \mid a < x < b \}, </math>  where {{mvar|a}} and {{mvar|b}} are real numbers

In the case of an open interval, <math>a</math> and <math>b</math> do not belong to <math>D</math>, and the values <math>f(a)</math> and <math>f(b)</math> are not defined, and if they are, they do not matter for continuity on <math>D</math>.

====Definition in terms of limits of functions====
The function {{math|''f''}} is ''continuous at some point'' {{math|''c''}} of its domain if the [[limit of a function|limit]] of <math>f(x),</math> as ''x'' approaches ''c'' through the domain of ''f'',  exists and is equal to <math>f(c).</math><ref>{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Undergraduate analysis | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=[[Undergraduate Texts in Mathematics]] | isbn=978-0-387-94841-6 | year=1997}}, section II.4</ref> In mathematical notation, this is written as
<math display="block">\lim_{x \to c}{f(x)} = f(c).</math>
In detail this means three conditions: first, {{math|''f''}} has to be defined at {{math|''c''}} (guaranteed by the requirement that {{math|''c''}} is in the domain of {{math|''f''}}). Second, the limit of that equation has to exist. Third, the value of this limit must equal <math>f(c).</math>

(Here, we have assumed that the domain of ''f'' does not have any [[isolated point]]s.)

====Definition in terms of neighborhoods====
A [[neighborhood (mathematics)|neighborhood]] of a point ''c'' is a set that contains, at least, all points within some fixed distance of ''c''. Intuitively, a function is continuous at a point ''c'' if the range of ''f'' over the neighborhood of ''c'' shrinks to a single point <math>f(c)</math> as the width of the neighborhood around ''c'' shrinks to zero. More precisely, a function ''f'' is continuous at a point ''c'' of its domain if, for any neighborhood <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>

As neighborhoods are defined in any [[topological space]], this definition of a continuous function applies not only for real functions but also when the domain and the [[codomain]] are [[topological space]]s and is thus the most general definition. It follows that a function is automatically continuous at every [[isolated point]] of its domain. For example, every real-valued function on the integers is continuous.

====Definition in terms of limits of sequences====
[[File:Continuity of the Exponential at 0.svg|thumb|The sequence {{math|exp(1/''n'')}} converges to {{math|1=exp(0) = 1}}]]
One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n \in \N}</math> of points in the domain which [[convergent sequence|converges]] to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math>  In mathematical notation, <math display="block">\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>

====Weierstrass and Jordan definitions (epsilon–delta) of continuous functions====
[[File:Example of continuous function.svg|right|thumb|Illustration of the {{mvar|ε}}-{{mvar|δ}}-definition: at {{math|1=''x'' = 2}}, any value {{math|δ ≤ 0.5}} satisfies the condition of the definition for {{math|1=''ε'' = 0.5}}.]]
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function <math>f : D \to \mathbb{R}</math> as above and an element <math>x_0</math> of the domain <math>D</math>, <math>f</math> is said to be continuous at the point <math>x_0</math> when the following holds: For any positive real number <math>\varepsilon > 0,</math> however small, there exists some positive real number <math>\delta > 0</math> such that for all <math>x</math> in the domain of <math>f</math> with <math>x_0 - \delta < x < x_0 + \delta,</math> the value of <math>f(x)</math> satisfies
<math display="block">f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.</math>

Alternatively written, continuity of <math>f : D \to \mathbb{R}</math> at <math>x_0 \in D</math> means that for every <math>\varepsilon > 0,</math> there exists a <math>\delta > 0</math> such that for all <math>x \in D</math>:
<math display="block">\left|x - x_0\right| < \delta ~~\text{ implies }~~ |f(x) - f(x_0)| < \varepsilon.</math>

More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small [[Topological neighborhood |neighborhood]] around <math>f\left(x_0\right),</math> we need to choose a small enough neighborhood for the <math>x</math> values around <math>x_0.</math> If we can do that no matter how small the <math>f(x_0)</math> neighborhood is, then <math>f</math> is continuous at <math>x_0.</math>

In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)|basis for the topology]], here the [[metric topology]].

Weierstrass had required that the interval <math>x_0 - \delta < x < x_0 + \delta</math> be entirely within the domain <math>D</math>, but Jordan removed that restriction.

====Definition in terms of control of the remainder====
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. 
A function <math>C: [0,\infty) \to [0,\infty]</math> is called a control function if
* ''C'' is non-decreasing
*<math>\inf_{\delta > 0} C(\delta) = 0</math>

A function <math>f : D \to R</math> is ''C''-continuous at <math>x_0</math> if there exists such a neighbourhood <math display="inline">N(x_0)</math> that 
<math display="block">|f(x) - f(x_0)| \leq C\left(\left|x - x_0\right|\right) \text{ for all } x \in D \cap N(x_0)</math>

A function is continuous in <math>x_0</math> if it is ''C''-continuous for some control function ''C''.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions <math>\mathcal{C}</math> a function is {{nowrap|<math>\mathcal{C}</math>-continuous}} if it is {{nowrap|<math>C</math>-continuous}} for some <math>C \in \mathcal{C}.</math> For example, the [[Lipschitz continuity|Lipschitz]], the [[Hölder continuous function]]s of exponent {{mvar|α}} and the [[uniformly continuous function]]s below are defined by the set of control functions 
<math display="block">\mathcal{C}_{\mathrm{Lipschitz}} = \{C : C(\delta) = K|\delta| ,\  K > 0\}</math> 
<math display="block">\mathcal{C}_{\text{Hölder}-\alpha} = \{C : C(\delta) = K |\delta|^\alpha, \ K > 0\}</math>
<math display="block">\mathcal{C}_{\text{uniform cont.}} = \{C : C(0) = 0 \}</math>
respectively.

====Definition using oscillation====
[[File:Rapid Oscillation.svg|thumb|The failure of a function to be continuous at a point is quantified by its [[Oscillation (mathematics)|oscillation]].]]
Continuity can also be defined in terms of [[Oscillation (mathematics)|oscillation]]: a function ''f'' is continuous at a point <math>x_0</math> if and only if its oscillation at that point is zero;<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> A benefit of this definition is that it {{em|quantifies}} discontinuity: the oscillation gives how {{em|much}} the function is discontinuous at a point.

This definition is helpful in [[descriptive set theory]] to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than <math>\varepsilon</math> (hence a [[G-delta set|<math>G_{\delta}</math> set]]) – and gives a rapid proof of one direction of the [[Lebesgue integrability condition]].<ref>''[http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF Introduction to Real Analysis],'' updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177</ref>

The oscillation is equivalent to the <math>\varepsilon-\delta</math> definition by a simple re-arrangement and by using a limit ([[lim sup]], [[lim inf]]) to define oscillation: if (at a given point) for a given <math>\varepsilon_0</math> there is no <math>\delta</math> that satisfies the <math>\varepsilon-\delta</math> definition, then the oscillation is at least <math>\varepsilon_0,</math> and conversely if for every <math>\varepsilon</math> there is a desired <math>\delta,</math> the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a [[metric space]].

====Definition using the hyperreals====
[[Cauchy]] defined the continuity of a function in the following intuitive terms: an [[infinitesimal]] change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34). [[Non-standard analysis]] is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the [[hyperreal numbers]]. In nonstandard analysis, continuity can be defined as follows.

{{block indent|em=1.5|text=A real-valued function {{math|''f''}} is continuous at {{mvar|x}} if its natural extension to the hyperreals has the property that for all infinitesimal {{math|''dx''}}, <math>f(x + dx) - f(x)</math> is infinitesimal<ref>{{cite web| url=http://www.math.wisc.edu/~keisler/calc.html |title=Elementary Calculus|work=wisc.edu}}</ref>}}

(see [[microcontinuity]]).  In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to [[Augustin-Louis Cauchy]]'s definition of continuity.