A function that is in $L^1(\mathbb{R})$ but not $L^{\infty}(I)$ for any open interval $I$?

Question

A function that is in $L^1(\mathbb{R})$ but not $L^{\infty}(I)$ for any open interval $I$? This is an exercise from an old measure theory book I'm working through... I've been working trying to build stuff with intensely oscillating functions and log's and what not but can't come up with anything... Example appreciated!!

Answer 1

Let $f(x) = x^{-1/2}\chi_{(0,1)}.$ Note $f$ is in $L^1$ but not $L^\infty.$ Now consider

$$g(x) = \sum_{n=1}^{\infty}\frac{f(x-r_n)}{2^n},$$

where $r_1,r_2,\dots$ are the rationals.