Discontinuity point - Knowlpedia

Let $U\subseteq\mathbb R^n$ , let $f:U\to\mathbb R^m$ , and let $a\in U$ . The point $a$ is a discontinuity point of $f$ if $f$ is not continuous at $a$ . Equivalently, there exists $\varepsilon>0$ such that for every $\delta>0$ there is $x\in U$ with

\lVert x-a\rVert<\delta \quad\text{and}\quad \lVert f(x)-f(a)\rVert\ge\varepsilon.

Remarks

In many common situations, discontinuity at $a$ can be detected by limits: if $\lim_{x\to a}f(x)$ exists in the sense of a limit at a point and is not equal to $f(a)$ , then $a$ is a discontinuity point. The collection of all such points is the set of discontinuities.

Examples

The sign function defined by $f(x)=1$ for $x>0$ , $f(x)=-1$ for $x<0$ , and $f(0)=0$ is discontinuous at $0$ .
The function $f:\mathbb R\to\mathbb R$ defined by $f(x)=1$ for rational $x$ and $f(x)=0$ for irrational $x$ is discontinuous at every real number.