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 X be a topological space. A real-valued function f:X→ℝ is cliquish at a point x∈X if for any ϵ>0 and any open neighborhood U of x there is a non-empty open set G⊂U such that

|f(y)−f(z)|<ϵ∀y,z∈G

Note that in the above definition, it is not necessary that x∈G.

Properties

• If f:X→ℝ is (quasi-)continuous then f is cliquish.
• If f:X→ℝ and g:X→ℝ are quasi-continuous, then f+g is cliquish.
• If f:X→ℝ is cliquish then f is the sum of two quasi-continuous functions .

Example

Consider the function f:ℝ→ℝ defined by f(x)=0 whenever x≤0 and f(x)=1 whenever 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 G⊂U such that y,z<0∀y,z∈G. Clearly this yields |f(y)−f(z)|=0∀y∈G thus f is cliquish.

In contrast, the function g:ℝ→ℝ defined by g(x)=0 whenever x is a rational number and g(x)=1 whenever x is an irrational number is nowhere cliquish, since every nonempty open set G contains some y1,y2 with |g(y1)−g(y2)|=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. http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record. 
• T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange 14 (2): 259–308. doi:10.2307/44151947.