Problem of multiple generality

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 $f:X\to \mathbb{ℝ}$ is quasi-continuous at a point $x\in X$ if for any every $ϵ>0$ and any open neighborhood $U$ of $x$ there is a non-empty open set $G\subset U$ such that

$|f(x)-f(y)|<ϵ\mathrm{\forall }y\in G$

Note that in the above definition, it is not necessary that $x\in G$.

Properties

• If $f:X\to \mathbb{ℝ}$ is continuous then $f$ is quasi-continuous
• If $f:X\to \mathbb{ℝ}$ is continuous and $g:X\to \mathbb{ℝ}$ is quasi-continuous, then $f+g$ is quasi-continuous.

Example

Consider the function $f:\mathbb{ℝ}\to \mathbb{ℝ}$ defined by $f(x)=0$ whenever $x\le 0$ 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 $G\subset U$ such that $y<0\mathrm{\forall }y\in G$. Clearly this yields $|f(0)-f(y)|=0\mathrm{\forall }y\in G$ thus f is quasi-continuous.

References

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