Editing Quasi-continuous function (section)

==Example==

Consider the function <math> f: \mathbb{R} \rightarrow \mathbb{R} </math> defined by <math> f(x) = 0 </math> whenever <math> x \leq 0 </math> and <math> f(x) = 1 </math> whenever <math> x > 0 </math>.  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 <math> G \subset U </math> such that <math> y < 0 \; \forall y \in G </math>.  Clearly this yields  <math> |f(0) - f(y)| = 0  \; \forall y \in G</math> thus f is quasi-continuous.

In contrast, the function  <math> g: \mathbb{R} \rightarrow \mathbb{R} </math> defined by <math> g(x) = 0 </math> whenever <math> x</math> is a rational number and <math> g(x) = 1 </math> whenever <math> x</math> is an irrational number is nowhere quasi-continuous, since every nonempty open set <math>G</math> contains some <math>y_1, y_2</math> with <math>|g(y_1) - g(y_2)| = 1</math>.