If $f_n \rightarrow f$ uniformly and $f$ is continuous is $f_n$ continuous for all $n$?

Question

As the problem title states, if $f_n \rightarrow f$ converges uniformly and $f$ is continuous is $f_n$ continuous for all $n$?

Answer 1

No, consider the sequence of functions $$f_n(x) = \begin{cases} 1/n & \text{if $x=0$} \\ 0 & \text{otherwise} \end{cases}$$ Then $f_n \to 0$ uniformly, and while $0$ is continuous, none of the $f_n$ are.

Answer 2

No. Simply knowing that a sequence (of anything) converges (in any sense) to a given limit tells us nothing about individual terms of the sequence. If $f_n\to f$ uniformly, then you can choose any integer $m$ and replace $f_m$ by any function you want, without changing the fact that the sequence converges to $f$.