# Series of functions, all sorts of convergence!

$$f_n$$ 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.

• Let $f_n$ be a function whose graph is an isosceles triangle with base $[0,1/n]$ and height $2n$. Then for fixed $x > 0$, for sufficiently large $n$ we have $\int_0^x f_n = 1$. But $f_n \to 0$ pointwise. – Bungo Oct 24 '18 at 2:37
• $F_n$ and $F$ are defined only up to a constant term. 1) cannot hold for all choices of these constants. – Kavi Rama Murthy Oct 24 '18 at 5:37