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Is it true that all differentiable square-integrable functions tend to zero at $\pm \infty$? If that is not true could you give a counterexample?

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  • $\begingroup$ Only if the limit exists. In which case the proof is trivial. $\endgroup$ Jul 19, 2019 at 9:18
  • $\begingroup$ If $\lim_{x\to \infty }$ exist, then yes it's true. Otherwise, you can have that $f\in L^p$ (where $p\geq 1$) and $\limsup_{x\to \infty }f(x)=+\infty $. But it's nevertheless true that $\liminf_{x\to \infty }f=0$. $\endgroup$
    – Surb
    Jul 19, 2019 at 9:18
  • $\begingroup$ $$\sum_{n\geq 1}\exp\left[-n^4(x-n)^2\right]\in L^2$$ but for any $n\in\mathbb{N}^+$ we have $f(n)\geq 1$. $\endgroup$ Jul 19, 2019 at 9:38
  • $\begingroup$ Is this function differentiable? $\endgroup$ Jul 19, 2019 at 9:41

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Let $f =\sum\limits_{n=1}^{\infty} nI_{(n,n+\frac 1 {n^{4}})}$. Can you see that this is a counterexample?

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