Real function/Continuity in a Point/General/Definition

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

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

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