$f:[0,\infty) \to [0, \infty)$ is a continuous function such that $\int_{0}^\infty f(x)dx < \infty$, what can be said about $f(n)$ for $n$ large.

Question

This is a previous year question from a competitive examination:

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If $f:[0,\infty) \to [0, \infty)$ is a continuous function such that $\int_{0}^\infty f(x)dx < \infty$, then which of the following are true:

1. $\{f_n\}$ is a bounded sequence.
2. $f(n) \to 0 $ as $n \to \infty$.
3. $\sum_{n=1}^\infty f(n)$ is convergent.

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The answer given is None of the above.

I tried constructing some examples but none of them worked. Any ideas/suggestions are highly appreciated.