Does there function with no bound and $\int_{-\infty}^{\infty} |f(x)|dx < \infty$ Consider $f(x)$ with $\displaystyle \int_{-\infty}^{\infty}|f(x)|dx < \infty$. Does there exist some function , which doesn't has a bound and has this property? 
I thought no, but what about function which are near the zero and goes to infinite around zero? 

There is. Take $$f(x)=\frac{1}{\sqrt[]{x}}\mathbf{1}_{(0,1]}$$ This functions has no bound, but we have:
\begin{align}
\int_\mathbb{R}|f(x)|\,dx=\int^1_0 \frac{1}{\sqrt[]{x}}\,dx= 2<\infty
\end{align}
