Real function/Continuity in a Point/General/Definition

Continuous function

Let ${}D\subseteq \mathbb {R}$ be a subset,

$f\colon D\longrightarrow \mathbb {R}$

a function, and ${}x\in D$ a point. We say that ${}f$ is continuous in the point ${}x$, if for every ${}\epsilon >0$, there exists a ${}\delta >0$, such that for all ${}x'\in D$ fulfilling ${}\vert {x-x'}\vert \leq \delta$, the estimate ${}\vert {f(x)-f(x')}\vert \leq \epsilon$ holds. We say that ${}f$ continuous, if it is continuous in every point ${}x\in D$