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 $$ f:X \rightarrow \mathbb{R} $$ 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 $$|g(y_1) - g(y_2)| = 1$$.