$L^2$ but not $L^{\infty}$

Question

In R where dimensions $d=1$? I know that if $s \ge 0.5$, then $H^s$ is in $L^{\infty}$. So we can find a function that is in $L^2$ but not in $L^{\infty}$. I would like an example for such a function.

Answer 1

$\frac{1}{\sqrt[4]{x}}$ is in $L^2((0,1))$ but not $L^\infty((0,1)).$

Answer 2

Another example: $|\ln |x||^{1/2} \in L^2((0,1)) \setminus L^\infty((0,1))$