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 Calculus Theory of continuous functions