Statements on function with finite integral over $[0, \inf[$ 
Let $f: [0, \infty[\to[0, \infty[$ be a continuous function such that:
  $$\int_0^\infty f(x) dx < \infty$$
  Which of the following statements are true? 



*

*The sequence $\{f(n)\}_{n\in\mathbb{N}} $ is bounded. 

*$f(n) \to 0$ as $n\to \infty $

*The series $\sum_{n=1}^\infty is convergent.


Intuitively i feel each option is true.
For (a),if $f(n)$ is unbounded,since $f$ is non negative,the integral cannot be finite.
For (b),if $f(n)$ does not tend to $0$,then again integral cannot be finite.
For (c),since the series should take a value lesser than or equal to the integral of $f$,the series should be convergent(since partial sums must be bounded).
However the answer says none of the options are correct! Where am i going wrong?

Hint: Start considering the function
$$
f(x)=\sum_{n=1}^\infty n\chi_{[\,n-1/(2^nn),n+1/(2^nn)\,]}(x),
$$
How does the graph of $f$ looks like? The function $f$ is not continuous, but it can be slightly modified so that it is continuous (just “change the rectangles by triangles”). Why is it a counterexample?
[Here $\chi_A(x)$ denotes the characteristic function of the set $A$; i.e. it is the function whose value at $A$ is $1$ if $x\in A$, and $0$ otherwise.]
