If $f: \mathbb{R} \to \mathbb{R}$ a function such that $\forall x \geq0$, $f(x)\geq0$; $\forall b>0$, f is bounded and integrable in $[0,b]$ and $\int \limits_{0}^{\infty} f(x)dx$ converges... Then if $f$ is continuous in $[0,\infty]$ then $\lim_{x \to \infty} f(x) = 0$. I believe this to be false, the counterexample I was thinking of was of a function $f(x)=$ a trigonometric function, this way the limit would not exist and therefore it would not be zero, however I couldn't think of one that fulfilled all the conditions, particularly the one about the improper integral.
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There's a similar question here – bjorn93 Apr 28 '20 at 17:23
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Think of a function which has small triangles with height $1$ and base $1/n^2$. Then it's continuous, integrable because $1/n^2$ is summable, but its limit at infinity does not exist.
Botond
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Also, a function with given properties is not necessarily continuous. Think of the same function given by @Botond but have $f(n)=2$ for $n$ being natural. Since $\mathbb N$ is countable, nothing changes in terms of integrability, and all the given assumptions are held but this function is not continuous.
Sam
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