Quasi-continuous function

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous 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 quasi-continuous 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


 * $$ |f(x) - f(y)| < \epsilon \;\;\;\; \forall y \in G $$

Note that in the above definition, it is not necessary that $$ x \in G $$.

Properties

 * If $$ f: X \rightarrow \mathbb{R} $$ is continuous then $$ f$$ is quasi-continuous
 * If $$ f: X \rightarrow \mathbb{R} $$ is continuous and $$ g: X \rightarrow \mathbb{R} $$ is quasi-continuous, then $$ f+g $$ is quasi-continuous.

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 quasi-continuous 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 $$ y < 0 \; \forall y \in G $$. Clearly this yields $$ |f(0) - f(y)| = 0  \; \forall y \in G$$ thus f is quasi-continuous.

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 quasi-continuous, since every nonempty open set $$G$$ contains some $$y_1, y_2$$ with $$|g(y_1) - g(y_2)| = 1$$.