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

Let $X$ be a topological space. A real-valued function $f:X \rightarrow \mathbb{R}$ 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 $f: X \rightarrow \mathbb{R}$ is (quasi-)continuous then $f$ is cliquish.
If $f: X \rightarrow \mathbb{R}$ and $g: X \rightarrow \mathbb{R}$ are quasi-continuous, then $f+g$ is cliquish.
If $f: X \rightarrow \mathbb{R}$ is cliquish then $f$ is the sum of two quasi-continuous functions .

Example

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ 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 $g: \mathbb{R} \rightarrow \mathbb{R}$ 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

{{cite journal

| author = Ján Borsík

| date = 2007–2008

| title = Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity

| journal = Real Analysis Exchange

| volume = 33

| issue = 2

| pages = 339–350

| url = http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record

}}

{{cite journal

| jstor=44151947

| author=T. Neubrunn

| title=Quasi-continuity

| journal=Real Analysis Exchange

| volume=14

| number=2

| pages=259–308

| year=1988

| doi=10.2307/44151947

}}

Category:Calculus

Category:Theory of continuous functions