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.
Let X {\displaystyle X} be a topological space. A real-valued function f : X → R {\displaystyle f:X\rightarrow \mathbb {R} } is quasi-continuous at a point x ∈ X {\displaystyle x\in X} if for any ϵ > 0 {\displaystyle \epsilon >0} and any open neighborhood U {\displaystyle U} of x {\displaystyle x} there is a non-empty open set G ⊂ U {\displaystyle G\subset U} such that
Note that in the above definition, it is not necessary that x ∈ G {\displaystyle x\in G} .
Consider the function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } defined by f ( x ) = 0 {\displaystyle f(x)=0} whenever x ≤ 0 {\displaystyle x\leq 0} and f ( x ) = 1 {\displaystyle f(x)=1} whenever x > 0 {\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 G ⊂ U {\displaystyle G\subset U} such that y < 0 ∀ y ∈ G {\displaystyle y<0\;\forall y\in G} . Clearly this yields | f ( 0 ) − f ( y ) | = 0 ∀ y ∈ G {\displaystyle |f(0)-f(y)|=0\;\forall y\in G} thus f is quasi-continuous.
In contrast, the function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } defined by g ( x ) = 0 {\displaystyle g(x)=0} whenever x {\displaystyle x} is a rational number and g ( x ) = 1 {\displaystyle g(x)=1} whenever x {\displaystyle x} is an irrational number is nowhere quasi-continuous, since every nonempty open set G {\displaystyle G} contains some y 1 , y 2 {\displaystyle y_{1},y_{2}} with | g ( y 1 ) − g ( y 2 ) | = 1 {\displaystyle |g(y_{1})-g(y_{2})|=1} .