Cliquish function

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

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 ${\displaystyle x\in G}$.

Properties

• 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 .

Example

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 ${\displaystyle |g(y_{1})-g(y_{2})|=1}$.

References

• 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.