Cliquish function explained

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 \in X if for any \epsilon > 0 and any open neighborhood U of x there is a non-empty open set G \subset U such that|f(y) - f(z)| < \epsilon \forall y,z \in G

Note that in the above definition, it is not necessary that

x \in 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 \leq 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 \subset U such that y,z < 0 \forall y,z \in G . Clearly this yields |f(y) - f(z)| = 0 \forall y \in 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 y _{1,} y _{2} with |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 .
• 44151947 . T. Neubrunn . Quasi-continuity . Real Analysis Exchange . 14 . 2 . 259 - 308 . 1988 . 10.2307/44151947 .