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$f_{n}$ and $f$ are continuous functions and $f_{n}\rightarrow f$ pointwise. Which of the following are correct?

  1. $\int _{0}^{x}F_{n}\left( t\right) dt\rightarrow\int _{0}^{x}F\left( t\right) dt$

  2. $\int _{0}^{x}f_{n}\left( t\right) dt\rightarrow\int _{0}^{x}f\left( t\right) dt$

  3. $F_{n}'\left( x\right) \rightarrow f\left( x\right)$

where $F_{n}\left( x\right) =\int f_{n}\left( x\right) dx$ and $F\left( x\right) =\int f\left( x\right) dx$

I think the key to this problem is to deduce whether $f_{n}\rightarrow f$ uniformly or not. I got stucked in how to prove that.

Any counter-examples for 1 and 2?

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You need to precise the $F_n$'s by setting $F_n(x):=\int_{x_0}^xf_n$. The formula $\int f_n$ is only defined up to a constant. –  1015 Apr 6 '13 at 15:25
    
Your $t(x)$ should be $f(x)$, and then $3$ is trivially true. –  1015 Apr 6 '13 at 15:28
    
$f_n\not{\rightarrow}f$. uniformly. –  user45099 Apr 6 '13 at 15:28
    
@user57 The assumption is: $f_n\rightarrow f$ pointwise. And $3$ does not mention uniform convergence. –  1015 Apr 6 '13 at 15:29
    
No, sorry, I was addressing OP's last sentence. –  user45099 Apr 6 '13 at 15:30
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