# Real Function/Continuity in a point/Characterization/Fact

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. Then the following statements are equivalent.
1. ${\displaystyle {}f}$ is continuous in the point ${\displaystyle {}x}$.
2. For every convergent sequence ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ in ${\displaystyle {}D}$ with ${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}=x}$ also the image sequence ${\displaystyle {}{\left(f(x_{n})\right)}_{n\in \mathbb {N} }}$ is convergent with limit ${\displaystyle {}f(x)}$.