I apologize in advance for how basic this question is...
Let $j:V\rightarrow V/U$ be the ultrapower map where U is an ultrafilter on a set S, and $j(x)=[c_x]$. Now, let $f\in j(0)$. Then $f$ is undefined almost everywhere on S, but we're assuming $f$ is a function defined on all of S, so we conclude no such $f$ exists, and so $j(0)=0$. Okay, that I get. Now look at $j(1)$. By the same argument, $f\in j(1)$ iff $f=\emptyset$ a.e. So now my issue; this isn't unique! As long as it's 0 almost everywhere, I can give $f$ random values elsewhere. And in any case, it's a function on S, not $\{\emptyset\}$.
Thank you!