$f\in L^2(\mathbb{R})\Rightarrow f\to 0, x\to\pm\infty$?

Question

As the title already suggests: Let $f\in L^2(\mathbb{R})$. Does this imply that $$ f\to 0\text{ as }x\to\pm\infty? $$

Answer 1

The answer is no. Consider $$f(x)=\sum_{n=1}^\infty \chi_{[n-\frac 1 {n^2},n+\frac 1 {n^2} ]}(x)$$

With $\chi$ the charasteristic function. $f$ is in $L^2$ but $f$ does not have a limit at $\infty$. You can also find similar examples which create a continuous function.