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 $|g(y_{1})-g(y_{2})|=1$.

References

• Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange. 33 (2): 339–350.
• T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange. 14 (2): 259–308. JSTOR 44151947.