Real function/Continuity in a point/Characterization/Fact

Sequence criterion for continuity

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

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

a function and ${}x\in D$ a point. Then the following statements are equivalent.

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

Proof, Write another proof