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Math Help - limit of a sequence of functions in a metric space

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by aliceinwonderland View Post
    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.
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