Editing Continuous function (section)

=={{anchor|Continuous map (topology)}}Continuous functions between topological spaces==
<!--Linked from [[Preference (economics)]] and [[Continuity (topology)]]-->

Another, more abstract, notion of continuity is the continuity of functions between [[topological space]]s in which there generally is no formal notion of distance, as there is in the case of [[metric space]]s. A topological space is a set ''X'' together with a topology on ''X'', which is a set of [[subset]]s of ''X'' satisfying a few requirements with respect to their unions and intersections that generalize the properties of the [[open ball]]s in metric spaces while still allowing one to talk about the [[neighborhood (mathematics)| neighborhoods]] of a given point. The elements of a topology are called [[open subset]]s of ''X'' (with respect to the topology).

A function
<math display="block">f : X \to Y</math>
between two topological spaces ''X'' and ''Y'' is continuous if for every open set <math>V \subseteq Y,</math> the [[Image (mathematics)#Inverse image|inverse image]]
<math display="block">f^{-1}(V) = \{x \in X \; | \; f(x) \in V \}</math>
is an open subset of ''X''. That is, ''f'' is a function between the sets ''X'' and ''Y'' (not on the elements of the topology <math>T_X</math>), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''.

This is equivalent to the condition that the [[Image (mathematics)#Inverse image|preimages]] of the [[closed set]]s (which are the complements of the open subsets) in ''Y'' are closed in ''X''.

An extreme example: if a set ''X'' is given the [[discrete topology]] (in which every subset is open), all functions
<math display="block">f : X \to T</math>
to any topological space ''T'' are continuous. On the other hand, if ''X'' is equipped with the [[indiscrete topology]] (in which the only open subsets are the empty set and ''X'') and the space ''T'' set is at least [[T0 space|T<sub>0</sub>]], then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

=== Continuity at a point ===
[[File:continuity topology.svg|right|frame|Continuity at a point: For every neighborhood ''V'' of <math>f(x)</math>, there is a neighborhood ''U'' of ''x'' such that <math>f(U) \subseteq V</math>]]
The translation in the language of neighborhoods of the [[(ε, δ)-definition of limit|<math>(\varepsilon, \delta)</math>-definition of continuity]] leads to the following definition of the continuity at a point:
{{Quote frame|A function <math>f : X \to Y</math> is continuous at a point <math>x \in X</math> if and only if for any neighborhood {{mvar|V}} of <math>f(x)</math> in {{mvar|Y}}, there is a neighborhood {{mvar|U}} of <math>x</math> such that <math>f(U) \subseteq V.</math>}}

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using [[preimage]]s rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and <math>f^{-1}(V)</math> is the largest subset {{mvar|U}} of {{mvar|X}} such that <math>f(U) \subseteq V,</math> this definition may be simplified into:
{{Quote frame|A function <math>f : X \to Y</math> is continuous at a point <math>x\in X</math> if and only if <math>f^{-1}(V)</math> is a neighborhood of <math>x</math> for every neighborhood {{mvar|V}} of <math>f(x)</math> in {{mvar|Y}}.}}

As an open set is a set that is a neighborhood of all its points, a function <math>f : X \to Y</math> is continuous at every point of {{mvar|''X''}} if and only if it is a continuous function.

If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the [[neighborhood system]] of [[open ball]]s centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above <math>\varepsilon-\delta</math> definition of continuity in the context of metric spaces.  In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a [[Hausdorff space]], 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.

Given <math>x \in X,</math> a map <math>f : X \to Y</math> is continuous at <math>x</math> if and only if whenever <math>\mathcal{B}</math> is a filter on <math>X</math> that [[Convergent filter|converges]] to <math>x</math> in <math>X,</math> which is expressed by writing <math>\mathcal{B} \to x,</math> then necessarily <math>f(\mathcal{B}) \to f(x)</math> in <math>Y.</math> 
If <math>\mathcal{N}(x)</math> denotes the [[neighborhood filter]] at <math>x</math> then <math>f : X \to Y</math> is continuous at <math>x</math> if and only if <math>f(\mathcal{N}(x)) \to f(x)</math> in <math>Y.</math>{{sfn|Dugundji|1966|pp=211–221}} Moreover, this happens if and only if the [[prefilter]] <math>f(\mathcal{N}(x))</math> is a [[filter base]] for the neighborhood filter of <math>f(x)</math> in <math>Y.</math>{{sfn|Dugundji|1966|pp=211–221}}

=== Alternative definitions ===
Several [[Characterizations of the category of topological spaces|equivalent definitions for a topological structure]] exist; thus, several equivalent ways exist to define a continuous function.

==== Sequences and nets {{anchor|Heine definition of continuity}}====
In several contexts, the topology of a space is conveniently specified in terms of [[limit points]]. This is often accomplished by specifying when a point is the [[limit of a sequence]]. Still, 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 family|indexed]] by a [[directed set]], known as [[Net (mathematics)|nets]]. A function is (Heine-)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 <math>f : X \to Y</math> is '''[[Sequential continuity|sequentially continuous]]''' if whenever a sequence <math>\left(x_n\right)</math> in <math>X</math> converges to a limit <math>x,</math> the sequence <math>\left(f\left(x_n\right)\right)</math> converges to <math>f(x).</math>  Thus, sequentially continuous functions "preserve sequential limits."  Every continuous function is sequentially continuous. If <math>X</math> is a [[first-countable space]] and [[Axiom of countable choice|countable choice]] holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if <math>X</math> 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 the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:<ref>{{cite book |title=Calculus and Analysis in Euclidean Space |edition=illustrated |first1=Jerry |last1=Shurman |publisher=Springer |year=2016 |isbn=978-3-319-49314-5 |pages=271–272 |url=https://books.google.com/books?id=wTmgDQAAQBAJ}}</ref>

{{math theorem|name=Theorem|note=|style=|math_statement=A function <math>f : A \subseteq \R \to \R</math> is continuous at <math>x_0</math> if and only if it is [[sequentially continuous]] at that point.
}}
{{collapse top|title=Proof|left=true}}
''Proof.'' Assume that <math>f : A \subseteq \R \to \R</math> is continuous at <math>x_0</math> (in the sense of [[(ε, δ)-definition of limit#Continuity|<math>\epsilon-\delta</math> continuity]]). Let <math>\left(x_n\right)_{n\geq1}</math> be a sequence converging at <math>x_0</math> (such a sequence always exists, for example, <math>x_n = x, \text{ for all } n</math>); since <math>f</math> is continuous at <math>x_0</math>
<math display=block>\forall \epsilon > 0\, \exists \delta_{\epsilon} > 0 : 0 < |x-x_0| < \delta_{\epsilon} \implies |f(x)-f(x_0)| < \epsilon.\quad (*)</math>
For any such <math>\delta_{\epsilon}</math> we can find a natural number <math>\nu_{\epsilon} > 0</math> such that for all <math>n > \nu_{\epsilon},</math>
<math display=block>|x_n-x_0| < \delta_{\epsilon},</math>
since <math>\left(x_n\right)</math> converges at <math>x_0</math>; combining this with <math>(*)</math> we obtain
<math display=block>\forall \epsilon > 0 \,\exists \nu_{\epsilon} > 0 : \forall n > \nu_{\epsilon} \quad |f(x_n)-f(x_0)| < \epsilon.</math>
Assume on the contrary that <math>f</math> is sequentially continuous and proceed by contradiction: suppose <math>f</math> is not continuous at <math>x_0</math>
<math display=block>\exists \epsilon > 0 : \forall \delta_{\epsilon} > 0,\,\exists x_{\delta_{\epsilon}}: 0 < |x_{\delta_{\epsilon}}-x_0| < \delta_\epsilon \implies |f(x_{\delta_{\epsilon}})-f(x_0)| > \epsilon</math>
then we can take <math>\delta_{\epsilon}=1/n,\,\forall n > 0</math> and call the corresponding point <math>x_{\delta_{\epsilon}} =: x_n</math>: in this way we have defined a sequence <math>(x_n)_{n\geq1}</math> such that
<math display=block>\forall n > 0 \quad |x_n-x_0| < \frac{1}{n},\quad |f(x_n)-f(x_0)| > \epsilon</math>
by construction <math>x_n \to x_0</math> but <math>f(x_n) \not\to f(x_0)</math>, which contradicts the hypothesis of sequential continuity. <math>\blacksquare</math>
{{collapse bottom}}

==== Closure operator and interior operator definitions ====

In terms of the [[Interior (topology)|interior]] and [[Closure (topology)|closure]] operators, we have the following equivalences,

{{math theorem|name=Theorem|note=|style=|math_statement=Let <math>f: X \to Y</math> be a mapping between topological spaces. Then the following are equivalent.
{{ordered list|type=lower-roman
  | <math>f</math> is continuous;
  | for every subset <math>B \subseteq Y,</math> <math>f^{-1}\left(\operatorname{int}_Y B\right) \subseteq \operatorname{int}_X\left(f^{-1}(B)\right);</math>
  | for every subset <math>A \subseteq X,</math> 
<math>f\left(\operatorname{cl}_X A\right) \subseteq \operatorname{cl}_Y \left(f(A)\right).</math>
}}

}}
{{collapse top|title=Proof|left=true}}
''Proof.''{{spaces|em}}'''i ⇒ ii'''.{{spaces|en}}
Fix a subset <math>B</math> of <math>Y.</math> Since <math>\operatorname{int}_Y B</math> is open.
and <math>f</math> is continuous, <math>f^{-1}(\operatorname{int}_Y B)</math> is open in <math>X.</math>
As <math>\operatorname{int}_Y B \subseteq B,</math> we have <math>f^{-1}(\operatorname{int}_Y B) \subseteq f^{-1}(B).</math>
By the definition of the interior, <math>\operatorname{int}_X\left(f^{-1}(B)\right)</math> is the largest open set contained in <math>f^{-1}(B).</math> Hence <math>f^{-1}(\operatorname{int}_Y B) \subseteq \operatorname{int}_X\left(f^{-1}(B)\right).</math>

'''ii ⇒ iii'''.{{spaces|en}}
Fix <math>A\subseteq X</math> and let <math>x\in\operatorname{cl}_X A.</math> Suppose to the contrary that <math>f(x)\notin\operatorname{cl}_Y\left(f(A)\right),</math>
then we may find some open neighbourhood <math>V</math> of <math>f(x)</math> that is disjoint from <math>\operatorname{cl}_Y\left(f(A)\right)</math>. By '''ii''', <math>f^{-1}(V) = f^{-1}(\operatorname{int}_Y V) \subseteq \operatorname{int}_X \left(f^{-1}(V)\right),</math> hence <math>f^{-1}(V)</math> is open. Then we have found an open neighbourhood of <math>x</math> that does not intersect <math>\operatorname{cl}_X A</math>, contradicting the fact that <math>x\in\operatorname{cl}_X A.</math>
Hence <math>f\left(\operatorname{cl}_X A\right) \subseteq \operatorname{cl}_Y \left(f(A)\right).</math>

'''iii ⇒ i'''.{{spaces|en}}
Let <math>N\subseteq Y</math> be closed. Let <math>M = f^{-1}(N)</math> be the preimage of <math>N.</math>
By '''iii''', we have <math>f\left(\operatorname{cl}_X M\right) \subseteq \operatorname{cl}_Y \left(f(M)\right).</math>
Since <math>f(M) = f(f^{-1}(N)) \subseteq N,</math>
we have further that <math>f\left(\operatorname{cl}_X M\right) \subseteq \operatorname{cl}_Y N = N.</math>
Thus <math>\operatorname{cl}_X M \subseteq f^{-1}\left(f(\operatorname{cl}_X M)\right) \subseteq f^{-1}(N) = M.</math>
Hence <math>M</math> is closed and we are done.
{{collapse bottom}}

If we declare that a point <math>x</math> is {{em|close to}} a subset <math>A \subseteq X</math> if <math>x \in \operatorname{cl}_X A,</math> then this terminology allows for a [[plain English]] description of continuity: <math>f</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math>f</math> maps points that are close to <math>A</math> to points that are close to <math>f(A).</math> Similarly, <math>f</math> is continuous at a fixed given point <math>x \in X</math> if and only if whenever <math>x</math> is close to a subset <math>A \subseteq X,</math> then <math>f(x)</math> is close to <math>f(A).</math>

Instead of specifying topological spaces by their [[Open set|open subsets]], any topology on <math>X</math> can [[Equivalence of categories|alternatively be determined]] by a [[Kuratowski closure operator|closure operator]] or by an [[interior operator]].  
Specifically, the map that sends a subset <math>A</math> of a topological space <math>X</math> to its [[Closure (topology)|topological closure]] <math>\operatorname{cl}_X A</math> satisfies the [[Kuratowski closure axioms]]. Conversely, for any [[Kuratowski closure operator|closure operator]] <math>A \mapsto \operatorname{cl} A</math> there exists a unique topology <math>\tau</math> on <math>X</math> (specifically, <math>\tau := \{ X \setminus \operatorname{cl} A : A \subseteq X \}</math>) such that for every subset <math>A \subseteq X,</math> <math>\operatorname{cl} A</math> is equal to the topological closure <math>\operatorname{cl}_{(X, \tau)} A</math> of <math>A</math> in <math>(X, \tau).</math> If the sets <math>X</math> and <math>Y</math> are each associated with closure operators (both denoted by <math>\operatorname{cl}</math>) then a map <math>f : X \to Y</math> is continuous if and only if <math>f(\operatorname{cl} A) \subseteq \operatorname{cl} (f(A))</math> for every subset <math>A \subseteq X.</math>

Similarly, the map that sends a subset <math>A</math> of <math>X</math> to its [[Interior (topology)|topological interior]] <math>\operatorname{int}_X A</math> defines an [[interior operator]]. Conversely, any interior operator <math>A \mapsto \operatorname{int} A</math> induces a unique topology <math>\tau</math> on <math>X</math> (specifically, <math>\tau := \{ \operatorname{int} A : A \subseteq X \}</math>) such that for every <math>A \subseteq X,</math> <math>\operatorname{int} A</math> is equal to the topological interior <math>\operatorname{int}_{(X, \tau)} A</math> of <math>A</math> in <math>(X, \tau).</math> If the sets <math>X</math> and <math>Y</math> are each associated with interior operators (both denoted by <math>\operatorname{int}</math>) then a map <math>f : X \to Y</math> is continuous if and only if <math>f^{-1}(\operatorname{int} B) \subseteq \operatorname{int}\left(f^{-1}(B)\right)</math> for every subset <math>B \subseteq Y.</math><ref>{{cite web|title=general topology - Continuity and interior|url=https://math.stackexchange.com/q/1209229|website=Mathematics Stack Exchange}}</ref>

==== Filters and prefilters ====
{{Main|Filters in topology}}

Continuity can also be characterized in terms of [[Filter (set theory)|filters]]. A function <math>f : X \to Y</math> is continuous if and only if whenever a filter <math>\mathcal{B}</math> on <math>X</math> [[Convergent filter|converges]] in <math>X</math> to a point <math>x \in X,</math> then the [[prefilter]] <math>f(\mathcal{B})</math> converges in <math>Y</math> to <math>f(x).</math> This characterization remains true if the word "filter" is replaced by "prefilter."{{sfn|Dugundji|1966|pp=211–221}}

===Properties===
If <math>f : X \to Y</math> and <math>g : Y \to Z</math> are continuous, then so is the composition <math>g \circ f : X \to Z.</math> If <math>f : X \to Y</math> 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 possible topologies on a fixed set ''X'' are [[partial ordering|partially ordered]]: a topology <math>\tau_1</math> is said to be [[Comparison of topologies|coarser]] than another topology <math>\tau_2</math> (notation: <math>\tau_1 \subseteq \tau_2</math>) if every open subset with respect to <math>\tau_1</math> is also open with respect to <math>\tau_2.</math> Then, the [[identity function|identity map]]
<math display="block">\operatorname{id}_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)</math>
is continuous if and only if <math>\tau_1 \subseteq \tau_2</math> (see also [[comparison of topologies]]). More generally, a continuous function
<math display="block">\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)</math>
stays continuous if the topology <math>\tau_Y</math> is replaced by a [[Comparison of topologies|coarser topology]] and/or <math>\tau_X</math> is replaced by a [[Comparison of topologies|finer topology]].

===Homeomorphisms===
Symmetric to the concept of a continuous map is an [[open map]], for which {{em|images}} of open sets are open. If an open map ''f'' has an [[inverse function]], that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a [[bijective]] function ''f'' between two topological spaces, the inverse function <math>f^{-1}</math> need not be continuous. A bijective continuous function with a continuous inverse function is called a {{em|[[homeomorphism]]}}.

If a continuous bijection has as its [[Domain of a function|domain]] a [[compact space]] and its codomain is [[Hausdorff space|Hausdorff]], then it is a homeomorphism.

===Defining topologies via continuous functions===
Given a function
<math display="block">f : X \to S,</math>
where ''X'' is a topological space and ''S'' is a set (without a specified topology), the [[final topology]] on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which <math>f^{-1}(A)</math> 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 is the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is [[surjective]], this topology is canonically identified with the [[quotient topology]] under the [[equivalence relation]] defined by ''f''.

Dually, for a function ''f'' from a set ''S'' to a topological space ''X'', the [[initial topology]] on ''S'' is defined by designating as an open set every subset ''A'' of ''S'' such that <math>A = f^{-1}(U)</math> for some open subset ''U'' of ''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 is the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the [[subspace topology]] of ''S'', viewed as a subset of ''X''.

A topology on a set ''S'' is uniquely determined by the class of all continuous functions <math>S \to X</math> into all topological spaces ''X''. [[Duality (mathematics)|Dually]], a similar idea can be applied to maps <math>X \to S.</math>