real analysis - If $f$ is continuous with $ int_0^{infty}f(t),dt

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.