$f:[0,\infty) \to [0,\infty)$ continuous function such that $\int_0^{\infty} f(x) dx$ is convergent ; is $\{f(n)\}$ bounded?

Question

Let $f:[0,\infty) \to [0,\infty)$ be a continuous function such that $\int_0^{\infty} f(x) dx$ is convergent ; is $\{f(n)\}$ bounded ? I know that if $f$ is uniformly continuous then $\{f(n)\}$ converges to $0$ , but I don't know what happens under assumption of continuity of $f$ only . Please help . Thanks in advance

Answer 1

Not necessarily. For example $f$ could be zero everywhere, except for having triangular spikes of height $n$ and width $n^{-3}$ centered around each $n$.

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If you want your $f$ to be $\mathcal C^{\infty}$, replace the triangular spikes with scaled and squashed copies of a fixed bump function.

You can even make it real analytic: $$ f(x) = \sum_{n=1}^{\infty} ne^{-(n^3(x-n))^2} $$