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

Consider the function defined by whenever and whenever. Clearly f is continuous everywhere except at x=0, thus quasi-continuous everywhere except 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.