If all functions $f_{n}$ are continuous in the interval  $[a,\,b]$  and  $\lim_{{n\to\infty}}f_{n}(x)=f(x)$  in all points $x$ of the interval, the limit function needs not to be continuous in this interval; example  $f_{n}(x)=\sin^{{n}}x$  in  $[0,\,\pi]$:

Note.  The notion of uniform convergence can be extended to the sequences of complex functions (the interval is replaced with some subset $G$ of $\mathbb{C}$).  The limit function of a uniformly convergent sequence of continuous functions is continuous in $G$.