function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$

Question

I'm searching for a function $f\notin L^{\infty}(\Omega)$, but $f\in L^p(\Omega)$ for all $1\le p <\infty$. And it has to be $|\Omega |<\infty$. I tried $f(x)=\frac{1}{2\sqrt{x}}$ and $\Omega= (0,1)$ and something like that, but it is $f\notin L^p(\Omega)$ for all $1\le p <\infty$. Could you help me?