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 be a topological space. A real-valued function is quasi-continuous at a point if for any and any open neighborhood of there is a non-empty open set such that

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

Properties

  • If is continuous then is quasi-continuous
  • If is continuous and is quasi-continuous, then is quasi-continuous.

Example

Consider the function defined by whenever and whenever . 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 such that . Clearly this yields thus f is quasi-continuous.

In contrast, the function defined by whenever is a rational number and whenever is an irrational number is nowhere quasi-continuous, since every nonempty open set contains some with .

References

  • Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange. 33 (2): 339–350.
  • T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange. 14 (2): 259–308. JSTOR 44151947.
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