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$f_n(x):[0,1]\rightarrow [0,1]$ are continuous and converges to $f(x)$, then

  1. $f$ is continuous.

  2. Convergence is uniform on $[0,1]$

  3. Convergence is uniform on $(0,1)$

  4. None of above statement is true.

Well, for 1 take $f_n(x)=x^n$, $f(x)=0$ for $ x \in [0,1)$ and $f(x)=1$ at $x=1$ hence 1 is false. For 2 I can give the same counter example as of 1. I have no idea about 3. Please help.

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    How do you show that your example works for 2? What your argument is will determine whether there is significant work left to be done to show that it works for $3$.2012-12-14

1 Answers 1

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HINT:

You can use the same counterexample with a little work for 3 too.

  • 1
    Be a bit more creative: make the point of discontinuity be, say, at $x=1/2$.2012-12-14