Real function/Continuity in a Point/General/Definition

Continuous function

Let $D\subseteq ℝ$ be a subset,

$f:D⟶ℝ$

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