Is sequential criterion of continuity holds for Sequence of continuous functions?

Question

I know that, If $(f_n)$ is Sequence of continuous functions on interval $I$ that converges uniformly on $I$ to function $f$ and If $(x_n)\subset I$ converges to $x_0\in I$ then,

$$lim(f_n(x_n))=f(x_0)$$

My question: Is the other direction holds? I mean, is the following is true?

Let $(f_n)$ be Sequence of continuous functions on $I$ and If for $c\in I$ we have, for every Sequence $(x_n)$ in $I$ that conveges to $c$, $lim(f_n(x_n))=f(c)$ then, Sequence $(f_n)$ converges uniformly to $f$ on $I$?

Please help...

Answer 1

Not true on the interval $(-\infty,\infty)$: let $$f_0(x) = \begin{cases} 0 & x<0, \\ x & 0\le x<1,\\ 1 & x\ge 1\end{cases}$$ And then set $f_n(x) := f_0(x-n)$. We have $f_n \to 0=:f$ pointwise, but not uniformly. Yet if $x_n\to c$, then $x_n$ is bounded, so there is some $M$ where $x_n < M$ for all $n$. Then for all $n>M$, $f_n(x_n)=0= f(c)$.

The same example works on $[0,\infty)$.