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.
DefinitionEdit
Let <math> X </math> be a topological space. A real-valued function <math> f:X \rightarrow \mathbb{R} </math> is quasi-continuous at a point <math> x \in X </math> if for any <math> \epsilon > 0 </math> and any open neighborhood <math> U </math> of <math> x </math> there is a non-empty open set <math> G \subset U </math> such that
- <math> |f(x) - f(y)| < \epsilon \;\;\;\; \forall y \in G </math>
Note that in the above definition, it is not necessary that <math> x \in G </math>.
PropertiesEdit
- If <math> f: X \rightarrow \mathbb{R} </math> is continuous then <math> f</math> is quasi-continuous
- If <math> f: X \rightarrow \mathbb{R} </math> is continuous and <math> g: X \rightarrow \mathbb{R} </math> is quasi-continuous, then <math> f+g </math> is quasi-continuous.
ExampleEdit
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>.