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Concept
Quasi-continuous function
Résumé
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.
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 .
If is continuous then is quasi-continuous
If is continuous and is quasi-continuous, then is quasi-continuous.
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 .
Source officielle
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