# Cliquish function

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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.

## Definition[edit]

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 .

## Properties[edit]

- If is (quasi-)continuous then is cliquish.
- If and are quasi-continuous, then is cliquish.
- If is cliquish then is the sum of two quasi-continuous functions .

## Example[edit]

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 .

## References[edit]

- Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity".
*Real Analysis Exchange*.**33**(2): 339–350. - T. Neubrunn (1988). "Quasi-continuity".
*Real Analysis Exchange*.**14**(2): 259–308. doi:10.2307/44151947. JSTOR 44151947.

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