A pointwise proposition for $L^2(\mathbb{R}^3)$ functions

Question

Suppose $f\in L^2(\mathbb{R}^3)$, do we have that

$f(x)\rightarrow 0$ as $|x|\rightarrow \infty$?

If this proposition holds, how to prove it? If not, pls give a counter example.

Answer 1

No. Consider $$ f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x \in \mathbb{Q}^3 \\ 0 & \mbox{otherwise.} \end{array} \right. $$