Series of functions, all sorts of convergence!

Question

$f_n$ and $f$ are continuous functions, and $f_n(x) \to f(x)$ pointwise. Which of the following is/are correct?

1. $\int_0^x F_n(t) \,dt \to \int_0^x F(t)\,dt$
2. $F_n'(x) \to f(x)$
3. $\int_0^x f_n(t)\,dt \to \int_0^x f(t)\,dt$

Here, $F(x) = \int f(x)\,dx$ and $F_n(x) = \int f_n(x)\,dx$.

My work. Clearly (2) is correct. But I am not sure about the others. Any help would be well-appreciated.

Answer 1

First and third are wrong, observe that for the integrals to converge, one requires uniform convergence and not pointwise. So the third is not necessarily true, whereas, if The integrals themselves don't, the integral of the integrals need not be equal, so first one is wrong !