# limit of a sequence of functions in a metric space

Give an example of a sequence $\{f_{n}\} , n=1,2, ,, \infty$ of continuous functions $f_{n}:I \rightarrow I$ such that for each x in I = [0,1], $\{f_{n}\} , n=1,2, ,, \infty$ converges to a real number f(x), but the limit function f is not continuous.
Give an example of a sequence ${f_{n}} n=1,2, ,, \infty$ of continuous functions $f_{n}:I \rightarrow I$ such that for each x in I = [0,1], ${f_{n}} n=1,2, ,, \infty$ converges to a real number f(x), but the limit function f is not continuous.
Consider $f_n (x) = x^n$ notice that $f_n : I\to I$ but the limit function $f(x) = \left\{ \begin{array}{c} 0 \text{ for }0\leq x < 1 \\ 1 \text{ for }x=1 \end{array}\right.$