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]\displaystyle{ X }[/math] be a topological space. A real-valued function [math]\displaystyle{ f:X \rightarrow \mathbb{R} }[/math] is quasi-continuous at a point [math]\displaystyle{ x \in X }[/math] if for any [math]\displaystyle{ \epsilon \gt 0 }[/math] and any open neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x }[/math] there is a non-empty open set [math]\displaystyle{ G \subset U }[/math] such that
- [math]\displaystyle{ |f(x) - f(y)| \lt \epsilon \;\;\;\; \forall y \in G }[/math]
Note that in the above definition, it is not necessary that [math]\displaystyle{ x \in G }[/math].
Properties
- If [math]\displaystyle{ f: X \rightarrow \mathbb{R} }[/math] is continuous then [math]\displaystyle{ f }[/math] is quasi-continuous
- If [math]\displaystyle{ f: X \rightarrow \mathbb{R} }[/math] is continuous and [math]\displaystyle{ g: X \rightarrow \mathbb{R} }[/math] is quasi-continuous, then [math]\displaystyle{ f+g }[/math] is quasi-continuous.
Example
Consider the function [math]\displaystyle{ f: \mathbb{R} \rightarrow \mathbb{R} }[/math] defined by [math]\displaystyle{ f(x) = 0 }[/math] whenever [math]\displaystyle{ x \leq 0 }[/math] and [math]\displaystyle{ f(x) = 1 }[/math] whenever [math]\displaystyle{ x \gt 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]\displaystyle{ G \subset U }[/math] such that [math]\displaystyle{ y \lt 0 \; \forall y \in G }[/math]. Clearly this yields [math]\displaystyle{ |f(0) - f(y)| = 0 \; \forall y \in G }[/math] thus f is quasi-continuous.
In contrast, the function [math]\displaystyle{ g: \mathbb{R} \rightarrow \mathbb{R} }[/math] defined by [math]\displaystyle{ g(x) = 0 }[/math] whenever [math]\displaystyle{ x }[/math] is a rational number and [math]\displaystyle{ g(x) = 1 }[/math] whenever [math]\displaystyle{ x }[/math] is an irrational number is nowhere quasi-continuous, since every nonempty open set [math]\displaystyle{ G }[/math] contains some [math]\displaystyle{ y_1, y_2 }[/math] with [math]\displaystyle{ |g(y_1) - g(y_2)| = 1 }[/math].
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.
 |  |
