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 ${\displaystyle X}$ be a topological space. A real-valued function ${\displaystyle f:X\rightarrow \mathbb {R} }$ is quasi-continuous at a point ${\displaystyle x\in X}$ if for any ${\displaystyle \epsilon >0}$ and any open neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ there is a non-empty open set ${\displaystyle G\subset U}$ such that

${\displaystyle |f(x)-f(y)|<\epsilon \;\;\;\;\forall y\in G}$

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

Properties

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

Example

Consider the function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ defined by ${\displaystyle f(x)=0}$ whenever ${\displaystyle x\leq 0}$ and ${\displaystyle f(x)=1}$ whenever ${\displaystyle x>0}$. 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 ${\displaystyle G\subset U}$ such that ${\displaystyle y<0\;\forall y\in G}$. Clearly this yields ${\displaystyle |f(0)-f(y)|=0\;\forall y\in G}$ 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.