Cliquish function

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) The topic of this article may not meet Wikipedia's general notability guideline. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted. Find sources: "Cliquish function" – news · newspapers · books · scholar · JSTOR (July 2024) (Learn how and when to remove this message) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Cliquish function" – news · newspapers · books · scholar · JSTOR (July 2024) (Learn how and when to remove this message) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (July 2024) (Learn how and when to remove this message) (Learn how and when to remove this message)

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 ${\displaystyle X}$ be a topological space. A real-valued function ${\displaystyle f:X\rightarrow \mathbb {R} }$ is cliquish at a point ${\displaystyle x\in X}$ if for any ${\displaystyle \epsilon >0}$ and any open neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ there is a non-empty open set ${\displaystyle G\subset U}$ such that

${\displaystyle |f(y)-f(z)|<\epsilon \;\;\;\;\forall y,z\in G}$

Note that in the above definition, it is not necessary that ${\displaystyle x\in G}$.

• If ${\displaystyle f:X\rightarrow \mathbb {R} }$ is (quasi-)continuous then ${\displaystyle f}$ is cliquish.
• If ${\displaystyle f:X\rightarrow \mathbb {R} }$ and ${\displaystyle g:X\rightarrow \mathbb {R} }$ are quasi-continuous, then ${\displaystyle f+g}$ is cliquish.
• If ${\displaystyle f:X\rightarrow \mathbb {R} }$ is cliquish then ${\displaystyle f}$ is the sum of two quasi-continuous functions .

Consider the function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ defined by ${\displaystyle f(x)=0}$ whenever ${\displaystyle x\leq 0}$ and ${\displaystyle f(x)=1}$ whenever ${\displaystyle x>0}$. 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 ${\displaystyle G\subset U}$ such that ${\displaystyle y,z<0\;\forall y,z\in G}$. Clearly this yields ${\displaystyle |f(y)-f(z)|=0\;\forall y\in G}$ thus f is cliquish.

In contrast, the function ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ defined by ${\displaystyle g(x)=0}$ whenever ${\displaystyle x}$ is a rational number and ${\displaystyle g(x)=1}$ whenever ${\displaystyle x}$ is an irrational number is nowhere cliquish, since every nonempty open set ${\displaystyle G}$ contains some ${\displaystyle y_{1},y_{2}}$ with ${\displaystyle |g(y_{1})-g(y_{2})|=1}$.

• 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.

This article needs additional or more specific categories. Please help out by adding categories to it so that it can be listed with similar articles. (July 2024)