Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
Cliquish function
Definition of 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.

We don't have any images related to Cliquish function yet.
We don't have any YouTube videos related to Cliquish function yet.
We don't have any PDF documents related to Cliquish function yet.
We don't have any Books related to Cliquish function yet.
We don't have any archived web articles related to Cliquish function yet.

Definition

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

| f ( y ) − f ( z ) | < ϵ ∀ y , z ∈ G {\displaystyle |f(y)-f(z)|<\epsilon \;\;\;\;\forall y,z\in G}

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

Properties

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

Example

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

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