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 .
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 .
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 .
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between and at some point within the interval.
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