$f_n$ and $f$ are continuous functions, and $f_n(x) \to f(x)$ pointwise. Which of the following is/are correct?
- $\int_0^x F_n(t) \,dt \to \int_0^x F(t)\,dt$
- $F_n'(x) \to f(x)$
- $\int_0^x f_n(t)\,dt \to \int_0^x f(t)\,dt$
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.