Answer whether the following statements are True or False.
Let $\left\{f_{n}\right\}_{n}$ be a sequence of (not necessarily continuous) functions from $[0,1]$ to $\mathbb{R}$. Let $f:[0,1] \rightarrow \mathbb{R}$ be such that for any $x \in[0,1]$ and any sequence $\left\{x_{n}\right\}_{n}$ consisting of elements from $[0,1]$, if $\displaystyle{}\lim _{n \rightarrow \infty} x_{n}=x$, then $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}\left(x_{n}\right)=f(x)$. Then $f$ is continuous.