# Square-integrable functions tend to zero at $\pm \infty$

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?

• Only if the limit exists. In which case the proof is trivial. Jul 19, 2019 at 9:18
• 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 $\liminf_{x\to \infty }f=0$.
– Surb
Jul 19, 2019 at 9:18
• $$\sum_{n\geq 1}\exp\left[-n^4(x-n)^2\right]\in L^2$$ but for any $n\in\mathbb{N}^+$ we have $f(n)\geq 1$. Jul 19, 2019 at 9:38
• Is this function differentiable? Jul 19, 2019 at 9:41

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