Editing Continuous function (section)

=== 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}}