Quasi-continuous function explained

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

X be a topological space. A real-valued function f:X → R is quasi-continuous at a point x \in X if for any \epsilon > 0 and any open neighborhood U of x there is a non-empty open set G \subset U such that|f(x) - f(y)| < \epsilon \forall y \in G

Note that in the above definition, it is not necessary that

x \in G .

Properties

• If

f: X → R is continuous then f is quasi-continuous

• If

f: X → R is continuous and g: X → R is quasi-continuous, then f + g is quasi-continuous.

Example

Consider the function

f: R → R defined by f(x) = 0 whenever x \leq 0 and f(x) = 1 whenever 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 \subset U such that y < 0 \forall y \in G . Clearly this yields |f( 0 ) - f(y)| = 0 \forall y \in G thus f is quasi-continuous.

In contrast, the function

g: R → R defined by g(x) = 0 whenever x is a rational number and g(x) = 1 whenever x is an irrational number is nowhere quasi-continuous, since every nonempty open set G contains some y _{1,} y _{2} with |g(y _{1)} - g(y _{2)|} = 1 .

See also

• Cliquish function

References

• Ján Borsík . 2007–2008 . Points of Continuity, Quasi-continuity, cliquishness, and Upper and Lower Quasi-continuity . Real Analysis Exchange . 33 . 2 . 339–350 .
• 44151947 . T. Neubrunn . Quasi-continuity . Real Analysis Exchange . 14 . 2 . 259 - 308 . 1988 .