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In mathematics, the notion of a cliquish function is similar to, but weaker than, the notion of a continuous function and quasi-continuous function. All (quasi-)continuous functions are cliquish but the converse is not true in general.
Let be a topological space. A real-valued function is cliquish 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 cliquish 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 cliquish.
In contrast, the function defined by whenever is a rational number and whenever is an irrational number is nowhere cliquish, since every nonempty open set contains some with .
This article needs additional or more specific
categories. (July 2024) |
![]() | This article has multiple issues. Please help
improve it or discuss these issues on the
talk page. (
Learn how and when to remove these template messages)
|
In mathematics, the notion of a cliquish function is similar to, but weaker than, the notion of a continuous function and quasi-continuous function. All (quasi-)continuous functions are cliquish but the converse is not true in general.
Let be a topological space. A real-valued function is cliquish 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 cliquish 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 cliquish.
In contrast, the function defined by whenever is a rational number and whenever is an irrational number is nowhere cliquish, since every nonempty open set contains some with .
This article needs additional or more specific
categories. (July 2024) |