Explain what "$f$ is continuous in $x_{0}$" means in terms of limits of sequences

Question

Task from an exam: Explain the meaning of "$f$ is continuous in $x_{0}$" in terms of limits of sequences.

I would explain it like that and please tell me if it's ok?

Let $f$ be a function which is continuous in $x_{0}$. The criteria "limits of sequences" (or however this is called... :o) says that there exists a sequence $x_{n}$ whose $\lim_{n\rightarrow\infty}x_{n}=x_{0}$

$\Rightarrow \lim_{n\rightarrow\infty}(f(x_{n}))=x_{0}=f(x_{0})$

Please tell me is this explanation correct?

Answer 1

I would say the following: let $\;f\;$ be a function defined on $\;x_0\;$ and some neighborhood $\;I_0\;$ of it. Then, $\;f\;$ is continuous at $\;x_0\;$ iff for any sequence $\;\{x_n\}\;$ in $\;I_0\;$ s.t.

$$\lim_{n\to\infty}x_n=x_0\;,\;\;\text{then it is true that also}\;\;\lim_{n\to\infty}f(x_n)=f(x_0)$$