Quasi-continuous function

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general.

## Contents

* 1 Definition
* 2 Properties
* 3 Example
* 4 See also
* 5 References

## Definition

Let X be a topological space. A real-valued function f:X->ℝ is quasi-continuous at a point x∈X if for any ϵ>0 and any open neighborhood U of x there is a non-empty open set G⊂U such that

|f(x)−f(y)|<ϵ∀y∈G

Note that in the above definition, it is not necessary that x∈G.

## Properties

* If f:X->ℝ is continuous then f is quasi-continuous
* If f:X->ℝ is continuous and g:X->ℝ is quasi-continuous, then f+g is quasi-continuous.

## Example

Consider the function f:ℝ->ℝ defined by f(x)=0 whenever x≤0 and f(x)=1 whenever x>0. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except (at most) at x=0. At x=0, take any open neighborhood U of x. Then there exists an open set G⊂U such that y<0∀y∈G. Clearly this yields |f(0)−f(y)|=0∀y∈G thus f is quasi-continuous.

In contrast, the function g:ℝ->ℝ defined by g(x)=0 whenever x is a rational number and g(x)=1 whenever x is an irrational number is nowhere quasi-continuous, since every nonempty open set G contains some y1,y2 with |g(y1)−g(y2)|=1.

## See also

* Cliquish function

## References

* Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange 33 (2): 339–350. http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record.
* T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange 14 (2): 259-308. doi:10.2307/44151947.

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