function vanishing at infinity and integrability

Question

Suppose $m_n$ is the Lebesgue measure on $\mathbb{R}^n$.

Def Let $f:\mathbb{R}^n\to \mathbb{R}$ be continuous. We say that $f$ vanishes at infinity if for every $\epsilon>0$ there is a compact $K\subseteq \mathbb{R}^n$ so that $|f|<\epsilon$ outside $K$.

Is it true that every continuous $f:\mathbb{R}^n\to \mathbb{R}$ that vanishes at infinity is integrable, ie $$\int_{\mathbb{R}^n}|f|dm_n<+\infty$$ ???

Thanks a lot in advance.

Answer 1

Try $1/(|x|+1)$ in the case $n=1$.