Let $$C_{0}(\mathbb R)= \big\{\,f:\mathbb R \to \mathbb C\,\,\, \text{continuous and}\,\, \lim_{x\to \pm \infty}f(x)=0 \big\}.$$
Assume that $f\in C_{0}(\mathbb R)$.
My question is: Is it always true that, $\int_{\mathbb R}|f(x)| dx < \infty ? $ If not, counter example ?
(My attempt: Since $f\in C_{0}(\mathbb R)$, so given $\epsilon >0$ there is a compact set $K\subset \mathbb R$ and $M> 0$ such that $|f(x)|\leq M$ for every $x\in K$ and $|f(x)|< \epsilon $ for every $x\in \mathbb R - K$; thus, $\int_{\mathbb R}|f(x)| dx = \int_{K}|f(x)| dx + \int_{\mathbb R -K} |f(x)|dx \leq C + \epsilon \mu(\mathbb R - K)$, where $\mu$ is Lebesgue measure on $\mathbb R$ ; but, certainly, this is incomplete argument !!)
