Let $f:[0,\infty)\to [0,\infty)$ be a continuous function such that $\displaystyle \int_0^{\infty}f(t)\,dt<\infty$. Which of the following statements are true ?
(A) The sequence $\{f(n)\}$ is bounded.
(B) $f(n)\to 0$ as $n \to \infty$.
(C) The series $\displaystyle \sum f(n)$ is convergent.
I am unable to prove directly but I am thinking about the function $f(x)=\frac{1}{1+x^2}$. For this function all options are correct. Is it correct ? I think not , as I have no proof in general.
Please help by giving a proof or disprove the statements.
Consider a function $f$ which is the sum of triangles, $\Gamma_n$ , where $\Gamma_n$ has base centered at $n$, having vertices at $(n+1/n^3,0)$, $(n-1/n^3,0)$ and $(n,n)$ i.e. function takes value $f(n)=n$ and outside the triangle $f$ takes value $0$, then, this function satisfies all hypothesis, and shows that all options are false.