$f_n$ 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.

  • 1
    $\begingroup$ 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. $\endgroup$ – Bungo Oct 24 '18 at 2:37
  • 1
    $\begingroup$ $F_n$ and $F$ are defined only up to a constant term. 1) cannot hold for all choices of these constants. $\endgroup$ – Kavi Rama Murthy Oct 24 '18 at 5:37

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 !


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