Rudin defined the essential range of a function in $L^{\infty}$ as the set $R_{f}$ of complex numbers $w$ such that $\mu(x:|f(x)-w|<\epsilon)>0,\forall \epsilon$
He asked the reader to prove that $R_{f}$ is compact. But since $f\in L^{\infty}$, we would have existence of functions like $f(x)=x$, whose $L^{\infty}$ norm is also infinity. Then any point in $\mathbb{R}$ could be in $R_{f}$, so $R_{f}$ cannot be bounded. I can show $R_{f}$ is closed, but I am not clear where I was wrong at here.