Editing Continuous function (section)

=== 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>
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==== 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.
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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}}