TITLE: Could non-continuous sequence of functions converge uniformly to continuous function?
QUESTION [8 upvotes]: If continuous sequence $ \left( f_n\left(x\right) \right)$ converges uniformly to function $f\left(x\right)$ in some interval of real numbers, than $f\left(x\right)$ must be also continuous.
But if non-continuous sequence $ \left( f_n\left(x\right) \right)$ converges uniformly to $f\left(x\right)$ , can $f\left(x\right)$ be continuous ?
Thanks.

REPLY [10 votes]: Yes, take
$$f_n(x)=\left\lbrace\begin{array}{cc}0&x\neq 0\\\frac{1}{n} & x=0\end{array} \right.$$