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In [[mathematics]], a '''continuous function''' is a [[function (mathematics)|function]] for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous [[inverse function]] is called "[[bicontinuous]]". An intuitive (though imprecise) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.

Continuity of functions is one of the core concepts of [[topology]], which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are [[real number]]s. In addition, this article discusses the definition for the more general case of functions between two [[metric space]]s. In [[order theory]], especially in [[domain theory]], one considers a notion of continuity known as [[Scott continuity]]. Other forms of continuity do exist but they are not discussed in this article.

As an example, consider the function ''h''(''t'') which describes the [[height]] of a growing flower at time ''t''. This function is continuous. In fact, there is a dictum of [[classical physics]] which states that ''in nature everything is continuous''. By contrast, if ''M''(''t'') denotes the amount of money in a bank account at time ''t'', then the function jumps whenever money is deposited or withdrawn, so the function ''M''(''t'') is discontinuous. (However, if one assumes a discrete set as the domain of function ''M'', for instance the set of points of time at 4:00 PM on business days, then ''M'' becomes continuous function, as every function whose domain is a discrete subset of reals is.)

== Real-valued continuous functions ==
===Historical infinitesimal definition===
[[Cauchy]] defined 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).

===Definition in terms of limits===
Suppose we have a function that maps [[real number]]s to real numbers and whose [[domain (mathematics)|domain]] is some [[interval (mathematics)|interval]], like the functions ''h'' and ''M'' above. Such a function can be represented by a [[graph of a function|graph]] in the [[Cartesian coordinate system|Cartesian plane]]; the function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] with no "holes" or "jumps".

In general, we say that the function ''f'' is continuous at some [[point (geometry)|point]] ''c'' of its domain if, and only if, the following holds:
* The [[limit of a function|limit]] of ''f''(''x'') as ''x'' approaches ''c'' through domain of ''f'' does exist and is equal to ''f''(''c''); in mathematical notation, <math>\lim_{x \to c}{f(x)} = f(c)</math>. If the point ''c'' in the domain of ''f'' is not a [[limit point]] of the domain, then this condition is [[vacuous truth|vacuously true]], since ''x'' cannot approach ''c'' through values not equal ''c''. Thus, for example, every function whose domain is the set of all integers is continuous.

We call a function '''continuous''' if and only if it is continuous at every point of its domain. More generally, we say that a function is continuous on some [[subset]] of its domain if it is continuous at every point of that subset.

The notation ''C''(Ω) or ''C''<sup>0</sup>(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, ''C''<sup>1</sup>(Ω) is used to denote the set of [[differentiable function]]s whose [[derivative]] is continuous, ''C''²(Ω) for the twice-differentiable functions whose [[second derivative]] is continuous, and so on (see [[differentiability class]]). In the field of computer graphics, these three levels are sometimes called ''g0'' (continuity of position), ''g1'' (continuity of tangency), and ''g2'' (continuity of curvature). The notation ''C''<sup>(''n'', α)</sup>(Ω) occurs in the definition of a more subtle concept, that of [[Hölder continuity]].

=== Weierstrass definition (epsilon-delta) of continuous functions===
Without resorting to limits, one can define continuity of real functions as follows.

Again consider a function ''ƒ'' that maps a set of [[real numbers]] to another set of real numbers, and suppose ''c'' is an element of the domain of ''ƒ''. The function ''ƒ'' is said to be continuous at the point ''c'' if the following holds: For any number ''ε''&nbsp;>&nbsp;0, however small, there exists some number ''δ''&nbsp;>&nbsp;0 such that for all ''x'' in the domain of ''ƒ'' with ''c''&nbsp;&minus;&nbsp;''δ''&nbsp;<&nbsp;''x''&nbsp;<&nbsp;''c''&nbsp;+&nbsp;''δ'', the value of ''ƒ''(''x'') satisfies

:<math> f(c) - \varepsilon < f(x) < f(c) + \varepsilon.\,</math>

Alternatively written: Given subsets ''I'', ''D'' of '''R''', continuity of ''ƒ''&nbsp;:&nbsp;''I''&nbsp;→&nbsp;''D'' at ''c''&nbsp;∈&nbsp;''I'' means that for every&nbsp;''ε''&nbsp;>&nbsp;0 there exists a ''δ''&nbsp;>&nbsp;0 such that for all ''x''&nbsp;∈&nbsp;''I'',:

:<math>| x - c | < \delta \Rightarrow | f(x) - f(c) | < \varepsilon. \, </math>

A form of this [[(ε, δ)-definition of limit|epsilon-delta definition]] of continuity was first given by [[Bernard Bolzano]] in 1817. Preliminary forms of a related definition of the limit were given by [[Augustin-Louis Cauchy|Cauchy]],<ref name="grabiner">{{Cite journal
|doi=10.2307/2975545
|title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
|first=Judith V.
|last=Grabiner
|journal=The [[American Mathematical Monthly]]
|month=March
|year=1983
|volume=90
|issue=3
|pages=185–194
|url=http://www.maa.org/pubs/Calc_articles/ma002.pdf
|postscript=.
|jstor=2975545
}}</ref> though the formal definition and the distinction between pointwise continuity and [[uniform continuity]] were first given by [[Karl Weierstrass]].

More intuitively, we can say that if we want to get all the ''ƒ''(''x'') values to stay in some small [[topological neighbourhood|neighborhood]] around ''ƒ''(''c''), we simply need to choose a small enough neighborhood for the ''x'' values around ''c'', and we can do that no matter how small the ''ƒ''(''x'') neighborhood is; ''ƒ'' is then continuous at&nbsp;''c''.

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]].

=== Heine definition of continuity ===
The following definition of continuity is due to [[Eduard Heine|Heine]].

:A real function ''&fnof;'' is continuous if for any [[sequence]] (''x''<sub>''n''</sub>) such that

::<math>\lim\limits_{n\to\infty} x_n=L,</math>

:it holds that

::<math>\lim\limits_{n\to\infty} f(x_n)=f(L).</math>

:(We assume that all the points ''x''<sub>''n''</sub> as well as ''L'' belong to the domain of ''&fnof;''.)

One can say, briefly, that a function is continuous if, and only if, it preserves limits.

Weierstrass's and Heine's definitions of continuity are equivalent on the reals. The usual (easier) proof makes use of the [[axiom of choice]], but in the case of global continuity of real functions it was proved by [[Wacław Sierpiński]] that the axiom of choice is not actually needed.<ref>{{cite web| title = Heine continuity implies Cauchy continuity without the Axiom of Choice|work=Apronus.com|url=http://www.apronus.com/math/cauchyheine.htm}}</ref>

In more general setting of topological spaces, the concept analogous to Heine definition of continuity is called ''sequential continuity''. In general, the condition of sequential continuity is [[Mathematical jargon#weaker|weaker]] than the analogue of Cauchy continuity, which is just called ''continuity'' (see [[continuity (topology)]] for details). However, if instead of sequences one uses ''[[Net (mathematics)|nets]]'' (sets indexed by a [[directed set]], not only the natural numbers), then the resulting concept is equivalent to the general notion of continuity in topology. Sequences are sufficient on metric spaces because they are [[first-countable space]]s (every point has a ''countable'' [[neighborhood basis]], hence representative points in each neighborhood are enough to ensure continuity), but general topological spaces are not first-countable, hence sequences do not suffice, and nets must be used.

=== 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 ƒ is continuous at a point ''x''<sub>0</sub> if and only if the oscillation 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 ''quantifies'' discontinuity: the oscillation gives how ''much'' the function is discontinuous at a point.

This definition is useful 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 ''ε'' (hence a [[G-delta set|G<sub>δ</sub> set]]) – and gives a very quick 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 equivalence to the ''ε''-''δ'' 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 ''ε''<sub>0</sub> there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''<sub>0</sub>, and conversely if for every ''ε'' there is a desired ''δ,'' 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===
[[Non-standard analysis]] is a way of making [[Isaac Newton|Newton]]-[[Gottfried Leibniz|Leibniz]]-style [[infinitesimals]] mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the [[hyperreal numbers]]. In nonstandard analysis, continuity can be defined as follows.

:A function ''&fnof;'' from the reals to the reals is continuous if its natural extension to the hyperreals has the property that for real ''x'' and infinitesimal ''dx'', {{nowrap|''&fnof;''(''x''+''dx'') &minus; ''&fnof;''(''x'')}} is infinitesimal.<ref>http://www.math.wisc.edu/~keisler/calc.html</ref>

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

=== Examples ===
* All [[polynomial|polynomial function]]s are continuous.
* If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions ''f''(''x'') = 1/''x'' and ''g''(''x'') = (sin&nbsp;''x'')/''x''. Neither function is defined at ''x'' = 0, so each has domain '''R''' \ {0} of [[real numbers]] except 0, and each function is continuous. The question of continuity at ''x'' = 0 does not arise, since ''x'' = 0 is neither in the domain of ''f'' nor in the domain of ''g''. The function ''f'' cannot be extended to a continuous function whose domain is '''R''', since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of ''g'' at 0 is 1, ''g'' can be extended continuously to '''R''' by defining its value at 0 to be 1.
* The [[exponential function]]s, [[logarithm]]s, [[square root]] function, [[trigonometric function]]s and [[absolute value]] function are continuous. [[Rational functions]], however, are not necessarily continuous on all of '''R'''.
* An example of a rational continuous function is ''f''(''x'')={{frac|1|x-2}}. The question of continuity at ''x''= 2 does not arise, since ''x'' = 2 is not in the domain of ''f''.
* An example of a discontinuous function is the function ''f'' defined by ''f''(''x'') = 1 if ''x'' > 0, ''f''(''x'') = 0 if ''x'' ≤ 0. Pick for instance ε = {{frac|1|2}}. There is no δ-neighborhood around ''x'' = 0 that will force all the ''f''(''x'') values to be within ε of ''f''(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values.
* Another example of a discontinuous function is the [[sign function|signum]] or sign function.
* A more complicated example of a discontinuous function is [[Thomae's function]].
* [[Dirichlet's function]]
::<math>D(x)=\begin{cases}
0\mbox{ if }x \in \mathbb{R} \setminus \mathbb{Q}\\
1\mbox{ if }x \in \mathbb{Q}
\end{cases}</math>
:is nowhere continuous.

=== Facts about continuous functions ===
If two functions ''f'' and ''g'' are continuous, then ''f'' + ''g'', ''fg'', and ''f''/''g'' are continuous. (Note. The only possible points ''x'' of discontinuity of ''f''/''g'' are the solutions of the equation ''g''(''x'') = 0; but then any such ''x'' does not belong to the domain of the function ''f''/''g''. Hence ''f''/''g'' is continuous on its entire domain, or - in other words - is continuous.)

The [[Function composition|composition]] ''f'' <small>o</small> ''g'' of two continuous functions is continuous.

If a function is [[differentiable]] at some point ''c'' of its domain, then it is also continuous at ''c''. The converse is not true: a function that is continuous at ''c'' need not be differentiable there. Consider for instance the [[absolute value]] function at ''c''&nbsp;=&nbsp;0.

====Intermediate value theorem====
The [[intermediate value theorem]] is an [[existence theorem]], based on the real number property of [[Real number#Completeness|completeness]], and states:

: If the real-valued function ''f'' is continuous on the [[interval (mathematics)|closed interval]] [''a'',&nbsp;''b''] and ''k'' is some number between ''f''(''a'') and ''f''(''b''), then there is some number ''c'' in [''a'',&nbsp;''b''] such that ''f''(''c'')&nbsp;=&nbsp;''k''.

For example, if a child grows from 1&nbsp;m to 1.5&nbsp;m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25&nbsp;m.

As a consequence, if ''f'' is continuous on [''a'',&nbsp;''b''] and ''f''(''a'') and ''f''(''b'') differ in [[Sign (mathematics)|sign]], then, at some point ''c'' in [''a'',&nbsp;''b''], ''f''(''c'') must equal [[0 (number)|zero]].

====Extreme value theorem====
The [[extreme value theorem]] states that if a function ''f'' is defined on a closed interval [''a'',''b''] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists ''c''&nbsp;∈&nbsp;[''a'',''b''] with ''f''(''c'') ≥ ''f''(''x'') for all ''x''&nbsp;∈&nbsp;[''a'',''b'']. The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (''a'',''b'') (or any set that is not both closed and bounded), as, for example, the continuous function ''f''(''x'') = 1/''x'', defined on the open interval (0,1), does not attain a maximum, being unbounded above.

===Directional continuity===
<div style="float:right;">
<gallery>Image:Right-continuous.svg|A right continuous function
Image:Left-continuous.svg|A left continuous function</gallery></div>
A function may happen to be continuous in only one direction, either from the "left" or from the "right". A '''right-continuous''' function is a function which is continuous at all points when approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows:

The function ''ƒ'' is said to be right-continuous at the point ''c'' if the following holds: For any number ''ε''&nbsp;&gt; 0 however small, there exists some number ''δ''&nbsp;&gt; 0 such that for all ''x'' in the domain with {{nowrap|''c'' &lt; ''x'' &lt; ''c'' + ''&delta;''}}, the value of ''ƒ''(''x'') will satisfy

:<math> |f(x) - f(c)| < \varepsilon.\,</math>

Notice that ''x'' must be larger than ''c'', that is on the right of ''c''. If ''x'' were also allowed to take values less than&nbsp;''c'', this would be the definition of continuity. This restriction makes it possible for the function to have a discontinuity at&nbsp;''c'', but still be right continuous at&nbsp;''c'', as pictured.

Likewise a '''left-continuous''' function is a function which is continuous at all points when approached from the left, that is, {{nowrap|''c'' &minus; ''&delta;'' &lt; ''x'' &lt; ''c''}}.

A function is continuous if and only if it is both right-continuous and left-continuous.

== Continuous functions between metric spaces ==
<!-- This section is linked from [[F-space]] -->
Now consider a function ''f'' from one [[metric space]] (''X'', d<sub>''X''</sub>) to another metric space (''Y'', d<sub>''Y''</sub>). Then ''f'' is continuous at the point ''c'' in ''X'' if for any positive real number ε, there exists a positive real number δ such that all ''x'' in ''X'' satisfying d<sub>''X''</sub>(''x'', ''c'') < δ will also satisfy d<sub>''Y''</sub>(''f''(''x''), ''f''(''c'')) < ε.

This can also be formulated in terms of [[sequence]]s and [[limit of a sequence|limits]]: the function ''f'' is continuous at the point ''c'' if for every sequence (''x''<sub>''n''</sub>) in ''X'' with limit lim ''x''<sub>''n''</sub> = ''c'', we have lim ''f''(''x''<sub>''n''</sub>) = ''f''(''c''). ''Continuous functions transform limits into limits.''

This latter condition can be weakened as follows: ''f'' is continuous at the point ''c'' if and only if for every convergent sequence (''x''<sub>''n''</sub>) in ''X'' with limit ''c'', the sequence (''f''(''x''<sub>''n''</sub>)) is a [[Cauchy sequence]], and ''c'' is in the domain of ''f''. ''Continuous functions transform convergent sequences into Cauchy sequences.''

The set of points at which a function between metric spaces is continuous is a [[Gδ set|G<sub>δ</sub> set]] – this follows from the ε-δ definition of continuity.

== Continuous functions between topological spaces ==
[[Image:continuity topology.svg|300px|right|frame|Continuity of a function at a point]]
The above definitions of continuous functions can be generalized to functions from one [[topological space]] to another in a natural way; a function ''f'' : ''X'' → ''Y'', where ''X'' and ''Y'' are topological spaces, is continuous [[if and only if]] for every [[open set]] ''V'' ⊆ ''Y'', the inverse image
:<math>f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}</math>
is open.

However, this definition is often difficult to use directly. Instead, suppose we have a function ''f'' from ''X'' to ''Y'', where ''X'', ''Y'' are topological spaces. We say ''f'' is '''continuous at ''x''''' for some ''x''&nbsp;∈&nbsp;''X'' if for any [[neighborhood (topology)|neighborhood]] ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'')&nbsp;⊆&nbsp;''V''.
Although this definition appears complex, the intuition is that no matter how "small" ''V'' becomes, we can always find a ''U'' containing ''x'' that will map inside it. If ''f'' is continuous at every ''x''&nbsp;∈&nbsp;''X'', then we simply say ''f'' is continuous.

In a [[metric space]], it is equivalent to consider the [[neighbourhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to ''x'' map to points close to ''f''(''x''). This only really makes sense in a metric space, however, which has a notion of distance.

Note, however, that if the target space is [[Hausdorff space|Hausdorff]], it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f(a)''. At an isolated point, every function is continuous.

=== Definitions ===
Several [[Characterizations of the category of topological spaces|equivalent definitions for a topological structure]] exist and thus there are several equivalent ways to define a continuous function.

==== Open and closed set definition ====
The most common notion of continuity in topology defines continuous functions as those functions for which the [[preimage]]s(or [[inverse image]]s) of [[open set]]s are open. Similar to the open set formulation is the '''closed set formulation''', which says that preimages (or inverse images) of [[closed set]]s are closed.

==== Neighborhood definition ====
Definitions based on preimages are often difficult to use directly. Instead, suppose we have a function {{nowrap|''f'' : ''X'' → ''Y''}}, where ''X'' and ''Y'' are topological spaces.<ref name="topological_function_in_depth">''f'' is a function {{nowrap|''f'' : ''X'' → ''Y''}} between two [[topological space]]s (''X'',''T<sub>X</sub>'') and (''Y'',''T<sub>Y</sub>''). That is, the function ''f'' is defined on the elements of the set ''X'', not on the elements of the topology ''T<sub>X</sub>''. However continuity of the function does depend on the topologies used.</ref> We say ''f'' is '''continuous at ''x''''' for some ''x''&nbsp;∈&nbsp;''X'' if for any [[neighborhood (topology)|neighborhood]] ''V'' of ''f''(''x''), there is a neighborhood ''U'' of ''x'' such that ''f''(''U'')&nbsp;⊆&nbsp;''V''.
Although this definition appears complicated, the intuition is that no matter how "small" ''V'' becomes, we can always find a ''U'' containing ''x'' that will map inside it. If ''f'' is continuous at every ''x''&nbsp;∈&nbsp;''X'', then we simply say ''f'' is continuous.
<center>[[Image:continuity topology.svg|300px|Continuity of a function at a point]]</center>

In a [[metric space]], it is equivalent to consider the [[neighbourhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This leads to the standard δ-ε definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to ''x'' map to points close to ''f''(''x''). This only really makes sense in a metric space, however, which has a notion of distance.

Note, however, that if the target space is [[Hausdorff space|Hausdorff]], it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f(a)''. At an isolated point, every function is continuous.

==== Sequences and nets ====
In several contexts, the topology of a space is conveniently specified in terms of [[limit points]]. In many instances, this is accomplished by specifying when a point is the [[limit of a sequence]], but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a [[directed set]], known as [[net (mathematics)|nets]]. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.

In detail, a function ''f'' : ''X'' → ''Y'' is '''sequentially continuous''' if whenever a sequence (''x''<sub>''n''</sub>) in ''X'' converges to a limit ''x'', the sequence (''f''(''x''<sub>''n''</sub>)) converges to ''f''(''x''). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If ''X'' is a [[first-countable space]], then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called [[sequential space]]s.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions.

==== Closure operator definition ====
Given two topological spaces (''X'',cl) and (''X''&nbsp;' ,cl&nbsp;') where cl and cl&nbsp;' are two [[closure operator]]s then a function
:<math>f:(X,\mathrm{cl}) \to (X' ,\mathrm{cl}')</math>
is '''continuous''' if for all subsets ''A'' of ''X''
:<math>f(\mathrm{cl}(A)) \subseteq \mathrm{cl}'(f(A)).</math>

One might therefore suspect that given two topological spaces (''X'',int) and (''X''&nbsp;' ,int&nbsp;') where int and int&nbsp;' are two [[interior operator]]s then a function
:<math>f:(X,\mathrm{int}) \to (X' ,\mathrm{int}')</math>
is '''continuous''' if for all subsets ''A'' of ''X''
:<math>f(\mathrm{int}(A)) \subseteq \mathrm{int}'(f(A))</math>
or perhaps if
:<math>f(\mathrm{int}(A)) \supseteq \mathrm{int}'(f(A));</math>
however, neither of these conditions is either necessary or sufficient for continuity.

Instead, we must resort to inverse images: given two topological spaces (''X'',int) and (''X''&nbsp;' ,int&nbsp;') where int and int&nbsp;' are two interior operators then a function
:<math>f:(X,\mathrm{int}) \to (X' ,\mathrm{int}')</math>
is '''continuous''' if for all subsets ''A'' of ''X'' '
:<math>f^{-1}(\mathrm{int}'(A)) \subseteq \mathrm{int}(f^{-1}(A)).</math>
We can also write that given two topological spaces (''X'',cl) and (''X''&nbsp;' ,cl&nbsp;') where cl and cl&nbsp;' are two closure operators then a function
:<math>f:(X,\mathrm{cl}) \to (X' ,\mathrm{cl}')</math>
is '''continuous''' if for all subsets ''A'' of ''X'' '
:<math>f^{-1}(\mathrm{cl}'(A)) \supseteq \mathrm{cl}(f^{-1}(A)).</math>

==== Closeness relation definition ====
Given two topological spaces (''X'',δ) and (''X''' ,δ') where δ and δ' are two [[closeness relation]]s then a function
:<math>f:(X,\delta) \to (X' ,\delta')</math>
is '''continuous''' if for all points ''x'' and of ''X'' and all subsets ''A'' of ''X'',
:<math>x \delta A \Rightarrow f(x)\delta'f(A).</math>
This is another way of writing the closure operator definition.

=== Useful properties of continuous maps ===
Some facts about continuous maps between topological spaces:
* If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'' : ''X'' → ''Z''.
* If ''f'' : ''X'' → ''Y'' is continuous and
** ''X'' is [[Compact space|compact]], then ''f''(''X'') is compact.
** ''X'' is [[Connected space|connected]], then ''f''(''X'') is connected.
** ''X'' is [[path-connected]], then ''f''(''X'') is path-connected.
** ''X'' is [[Lindelöf space|Lindelöf]], then ''f''(''X'') is Lindelöf.
** ''X'' is [[separable space|separable]], then ''f''(''X'') is separable.
*The [[identity function|identity map]] id<sub>X</sub> : (''X'', τ<sub>2</sub>) → (''X'', τ<sub>1</sub>) is continuous if and only if τ<sub>1</sub> ⊆ τ<sub>2</sub> (see also [[comparison of topologies]]).

=== Other notes ===
If a set is given the [[discrete topology]], all functions with that space as a domain are continuous. If the domain set is given the [[indiscrete topology]] and the range set is at least [[T0 space|T<sub>0</sub>]], then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Given a set ''X'', a [[partial ordering]] can be defined on the possible [[topology|topologies]] on ''X''. A continuous function between two topological spaces stays continuous if we [[stronger topology|strengthen]] the topology of the [[Domain of a function|domain space]] or [[weaker topology|weaken]] the topology of the [[codomain space]]. Thus we can consider the continuity of a given function a [[topological property]], depending only on the topologies of its domain and codomain spaces.

For a function ''f'' from a topological space ''X'' to a set ''S'', one defines the [[final topology]] on ''S'' by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''<sup>−1</sup>(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is [[Comparison of topologies|coarser]] than the final topology on ''S''. Thus the final topology can be characterized as the finest topology on ''S'' which makes ''f'' continuous. If ''f'' is [[surjective]], this topology is canonically identified with the [[quotient topology]] under the [[equivalence relation]] defined by ''f''. This construction can be generalized to an arbitrary family of functions ''X'' → ''S''.

Dually, for a function ''f'' from a set ''S'' to a topological space, one defines the [[initial topology]] on ''S'' by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which ''f''(''A'') is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus the initial topology can be characterized as the coarsest topology on ''S'' which makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the [[subspace topology]] of ''S'', viewed as a subset of ''X''. This construction can be generalized to an arbitrary family of functions ''S'' → ''X''.

Symmetric to the concept of a continuous map is an [[open map]], for which ''images'' of open sets are open. In fact, if an open map ''f'' has an inverse, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open.

If a function is a [[bijection]], then it has an [[inverse function]]. The inverse of a continuous bijection is open, but need not be continuous. If it is, this special function is called a [[homeomorphism]].
If a continuous bijection has as its domain a [[compact space]] and its codomain is [[Hausdorff space|Hausdorff]], then it is automatically a homeomorphism.

== Continuous functions between partially ordered sets ==
In [[order theory]], continuity of a function between [[Partially ordered sets|posets]] is [[Scott continuity]]. Let ''X'' be a [[complete lattice]], then a function ''f'' : ''X'' → ''X'' is continuous if, for each subset ''Y'' of ''X'', we have [[supremum|sup]]&nbsp;''f''(''Y'') = ''f''(sup&nbsp;''Y'').

== Continuous binary relation ==
A binary [[relation (mathematics)|relation]] ''R'' on ''A'' is continuous if ''R''(''a'', ''b'') whenever there are sequences (''a''<sup>''k''</sup>)<sub>''i''</sub> and (''b''<sup>''k''</sup>)<sub>''i''</sub> in ''A'' which converge to ''a'' and ''b'' respectively for which ''R''(''a''<sup>''k''</sup>,&nbsp;''b''<sup>''k''</sup>) for all ''k''. Clearly, if one treats ''R'' as a [[Indicator function|characteristic function]] in two variables, this definition of continuous is identical to that for continuous functions.

== Continuity space ==

A '''continuity space'''<ref>[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.851&rep=rep1&type=pdf Quantales and continuity spaces], RC Flagg - Algebra Universalis, 1997</ref><ref>All topologies come from generalized metrics, R Kopperman - American Mathematical Monthly, 1988</ref> is a generalization of metric spaces and posets, which uses the concept of [[quantale]]s, and that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.<ref>Continuity spaces: Reconciling domains and metric spaces, B Flagg, R Kopperman - Theoretical Computer Science, 1997</ref>

== See also ==
<div style="-moz-column-count:2; column-count:2;">
* [[Absolute continuity]]
* [[Bounded linear operator]]
* [[Classification of discontinuities]]
* [[Coarse function]]
* [[Continuous functor]]
* [[Continuous stochastic process]]
* [[Dini continuity]]
* [[Discrete function]]
* [[Equicontinuity]]
* [[Lipschitz continuity]]
* [[Normal function]]
* [[Piecewise]]
* [[Scott continuity]]
* [[Semicontinuity]]
* [[Smooth function]]
* [[Symmetrically continuous function]]
* [[Uniform continuity]]
</div>

==Notes==
{{commonscat|Continuity (functions)}}
<references/>

==References==
*[http://archives.math.utk.edu/visual.calculus/ Visual Calculus] by Lawrence S. Husch, [[University of Tennessee]] (2001)

[[Category:Calculus]]
[[Category:Continuous mappings| ]]
[[Category:Types of functions]]

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