Quasi-continuous function

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 ${\displaystyle x\in G}$.

• 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.

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 most) 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.

In contrast, the function ${\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} }$ defined by ${\displaystyle g(x)=0}$ whenever ${\displaystyle x}$ is a rational number and ${\displaystyle g(x)=1}$ whenever ${\displaystyle x}$ is an irrational number is nowhere quasi-continuous, since every nonempty open set ${\displaystyle G}$ contains some ${\displaystyle y_{1},y_{2}}$ with ${\displaystyle |g(y_{1})-g(y_{2})|=1}$.

• 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.