Metric spaces/Mapping/Continuity in a point/Definition

Let ${\displaystyle {}(L,d_{1})}$ and ${\displaystyle {}(M,d_{2})}$ be metric spaces,

${\displaystyle f\colon L\longrightarrow M}$

a mapping, and ${\displaystyle {}x\in L}$. The mapping ${\displaystyle {}f}$ is called continuous in ${\displaystyle {}x}$ if for every ${\displaystyle {}\epsilon >0}$ there exists a ${\displaystyle {}\delta >0}$ such that

${\displaystyle {}f{\left(B\left(x,\delta \right)\right)}\subseteq B\left(f(x),\epsilon \right)\,}$

holds. The mapping ${\displaystyle {}f}$ is called continuous if it is continuous in ${\displaystyle {}x}$ for every ${\displaystyle {}x\in L}$.