Editing Limit of a function (section)

==Functions on topological spaces==
{{See also|Filters in topology#Limits of functions}}

Suppose {{mvar|X}} and {{mvar|Y}} are [[topological space]]s with {{mvar|Y}} a [[Hausdorff space]]. Let {{mvar|p}} be a [[limit point]] of {{math|Ω ⊆ ''X''}}, and {{math|''L'' ∈ ''Y''}}. For a function {{math|''f'' : Ω → ''Y''}}, it is said that the '''limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} is {{mvar|L}}''', written

<math display=block> \lim_{x \to p}f(x) = L ,</math>

if the following property holds:

{{block indent|For every open [[Neighbourhood (mathematics)|neighborhood]] {{mvar|V}} of {{mvar|L}}, there exists an open neighborhood {{mvar|U}} of {{mvar|p}} such that {{math|''f''(''U'' ∩ &Omega; &minus; {''p''}) ⊆ ''V''}}.}}

This last part of the definition can also be phrased "there exists an open [[Neighbourhood (mathematics)#Deleted neighbourhood|punctured neighbourhood]] {{mvar|U}} of {{mvar|p}} such that {{math|''f''(''U'' ∩ Ω) ⊆ ''V''}}".

The domain of {{mvar|f}} does not need to contain {{mvar|p}}. If it does, then the value of {{mvar|f}} at {{mvar|p}} is irrelevant to the definition of the limit. In particular, if the domain of {{mvar|f}} is {{math|''X'' &minus; {''p''} }} (or all of {{mvar|X}}), then the limit of {{mvar|f}} as {{math|''x'' → ''p''}} exists and is equal to {{mvar|L}} if, for all subsets {{math|Ω}} of {{mvar|X}} with limit point {{mvar|p}}, the limit of the restriction of {{mvar|f}} to {{math|Ω}} exists and is equal to {{mvar|L}}. Sometimes this criterion is used to establish the ''non-existence'' of the two-sided limit of a function on {{tmath|\R}} by showing that the [[one-sided limit]]s either fail to exist or do not agree. Such a view is fundamental in the field of [[general topology]], where limits and continuity at a point are defined in terms of special families of subsets, called [[Filter (set theory)|filters]], or generalized sequences known as [[Net (mathematics)|nets]].

Alternatively, the requirement that {{mvar|Y}} be a Hausdorff space can be relaxed to the assumption that {{mvar|Y}} be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about ''the limit'' of a function at a point, but rather ''a limit'' or ''the set of limits'' at a point.

A function is continuous at a limit point {{mvar|p}} of and in its domain if and only if {{math|''f''(''p'')}} is ''the'' (or, in the general case, ''a'') limit of {{math|''f''(''x'')}} as {{mvar|x}} tends to {{mvar|p}}.

There is another type of limit of a function, namely the '''sequential limit'''. Let {{math|''f'' : ''X'' → ''Y''}} be a mapping from a topological space {{mvar|X}} into a Hausdorff space {{mvar|Y}}, {{math|''p'' ∈ ''X''}} a limit point of {{mvar|X}} and {{math|''L'' ∈ ''Y''}}. The sequential limit of {{mvar|f}} as {{mvar|x}} tends to {{mvar|p}} is {{mvar|L}} if
{{anchor|Heine definition of limit}}
:For every [[sequence (mathematics)|sequence]] ({{mvar|x<sub>n</sub>}}) in {{math|''X'' &minus; {''p''} }} that [[limit of a sequence|converges]] to {{mvar|p}}, the sequence {{math|''f''(''x''<sub>''n''</sub>)}} [[limit of a sequence|converges]] to {{mvar|L}}.

If {{mvar|L}} is the limit (in the sense above) of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}}, then it is a sequential limit as well, however the converse need not hold in general. If in addition {{mvar|X}} is [[metrizable]], then {{mvar|L}} is the sequential limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} if and only if it is the limit (in the sense above) of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}}.