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

Example

Consider the function $f: \mathbb \rightarrow \mathbb$ 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 $g: \mathbb \rightarrow \mathbb$ 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 $, g(y_1) - g(y_2), = 1$.

References

*
* {{cite journal
, jstor=44151947
, author=T. Neubrunn
, title=Quasi-continuity
, journal=Real Analysis Exchange
, volume=14
, number=2
, pages=259–308
, year=1988

Calculus
Theory of continuous functions