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.

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)| 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 . 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==

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