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 be a topological space. A real-valued function f : X → R is quasi-continuous at a point x ∈ X if for any every ϵ > 0 and any open neighborhood U of x there is a non-empty open set G ⊂ U such that
| f ( x ) − f ( y ) | < ϵ ∀ y ∈ G Note that in the above definition, it is not necessary that x ∈ G .
If f : X → R is continuous then f is quasi-continuousIf f : X → R is continuous and g : X → R is quasi-continuous, then f + g is quasi-continuous.Consider the function f : R → R defined by f ( x ) = 0 whenever x ≤ 0 and f ( x ) = 1 whenever 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 G ⊂ U such that y < 0 ∀ y ∈ G . Clearly this yields | f ( 0 ) − f ( y ) | = 0 ∀ y ∈ G thus f is quasi-continuous.
