Editing 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.

==Definition==

Let <math> X </math> be a [[topological space]].  A real-valued function <math> f:X \rightarrow \mathbb{R} </math> is quasi-continuous at a point <math> x \in X </math> if for any <math> \epsilon > 0 </math> and any [[open neighborhood]] <math> U </math> of <math> x </math> there is a non-empty [[open set]] <math> G \subset U </math> such that

: <math>  |f(x) - f(y)| < \epsilon \;\;\;\; \forall y \in G </math>

Note that in the above definition, it is not necessary that <math> x \in G </math>.

==Properties==

* If <math> f: X \rightarrow \mathbb{R} </math> is continuous then <math> f</math> is quasi-continuous
* If <math> f: X \rightarrow \mathbb{R} </math> is continuous and <math> g: X \rightarrow \mathbb{R} </math> is quasi-continuous, then <math> f+g </math> is quasi-continuous.

==Example==

Consider the function <math> f: \mathbb{R} \rightarrow \mathbb{R} </math> defined by <math> f(x) = 0 </math> whenever <math> x \leq 0 </math> and <math> f(x) = 1 </math> whenever <math> x > 0 </math>.  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 <math> G \subset U </math> such that <math> y < 0 \; \forall y \in G </math>.  Clearly this yields  <math> |f(0) - f(y)| = 0  \; \forall y \in G</math> thus f is quasi-continuous.

In contrast, the function  <math> g: \mathbb{R} \rightarrow \mathbb{R} </math> defined by <math> g(x) = 0 </math> whenever <math> x</math> is a rational number and <math> g(x) = 1 </math> whenever <math> x</math> is an irrational number is nowhere quasi-continuous, since every nonempty open set <math>G</math> contains some <math>y_1, y_2</math> with <math>|g(y_1) - g(y_2)| = 1</math>.

== See also ==
*[[Cliquish function]]

==References==

* {{cite journal
 | author = Ján Borsík
 | date = 2007–2008
 | title = Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity
 | journal = Real Analysis Exchange
 | volume = 33 
 | issue = 2
 | pages = 339–350
 | url = http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record
 }}
* {{cite journal
 | jstor=44151947
 |  author=T. Neubrunn
 | title=Quasi-continuity
 | journal=Real Analysis Exchange
 | volume=14
 | number=2
 | pages=259&ndash;308
 | year=1988 
 | doi=10.2307/44151947
 }} 

[[Category:Calculus]]
[[Category:Theory of continuous functions]]