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 every $\epsilon > 0$ 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 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.

References

• Jan Borsik (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange 33 (2): 339–350.