可积函数在∞趋于0

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Do integrable functions vanish at infinity?
可积函数在∞存在极限, 则为0.
但可积函数在∞不一定存在极限: 函数 $f:\mathbb R\to\mathbb R$ 在负半轴为0. 对每个$n\in\Bbb Z^+$,$$f(x)=\begin{cases}n&x\in\left[n,n+\frac{1}{n^3}\right]\\0&x\in\left(n+\frac{1}{n^3},n+1\right)\end{cases}$$则\[\int_{-\infty}^\infty f(x)\rmd x=\sum_{n=1}^\infty\frac1{n^2}<∞\]但是$f$在∞极限不存在

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Integral Transforms page 46
Exercise: think of an integrable function which does not satisfy this condition for any $p$. The point? It is not true that if a function is integrable then it must vanish at infinity. [Hint: think of narrow top-hats near integer values of $x$.]