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 ${\displaystyle f:X\to \mathbb{R}}$ is quasi-continuous at a point ${\displaystyle x\in X}$ if for any ${\displaystyle ϵ>0}$ and any open neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ there is a non-empty open set ${\displaystyle G\subset U}$ such that

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

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

Properties

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

Example

Consider the function ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$ defined by ${\displaystyle f(x)=0}$ whenever ${\displaystyle x\le 0}$ 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 ${\displaystyle y<0\mathrm{\forall }y\in G}$. Clearly this yields ${\displaystyle |f(0)-f(y)|=0\mathrm{\forall }y\in G}$ thus f is quasi-continuous.

In contrast, the function ${\displaystyle g:\mathbb{R}\to \mathbb{R}}$ 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 ${\displaystyle |g(y_1)-g(y_2)|=1}$.

References

• Ján Borsík (2007–2008). "Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity". Real Analysis Exchange 33 (2): 339–350. http://www.projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rae/1229619412&page=record. 
• T. Neubrunn (1988). "Quasi-continuity". Real Analysis Exchange 14 (2): 259–308. 

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Quasi-continuous function. Read more