My question is as follows
(1) is there discontinuous sequence of function $<f_n>$, uniformly converge to discontinuous function $f$?
(2) is there sequence of function $<f_n>$ which does not uniformly converge to any function(it doesn't matter that the function is continuous or discontinuous. what else!)
I tried to find those function. In my opinion, $f_n = x^n$ converge uniformly to function which value is zero except for $x$=1(but it does not converge uniformly to $f=0$. But, this example is not applicable to the question (1) because $f_n$ is continuous sequence.
1). I presume you mean a sequence of discontinuous functions, not a discontinuous sequence. Yes, just take any discontinuous function $f$ and let $f_n(x) = f(x) + 1/n$.
2) Any sequence of functions that fails to converge at some point, or converges but not uniformly.