If $f(x,y)=x$ , then is it true that $f$ is a closed map?

Question

If $f(x,y)=x$ , then is it true that $f$ is a closed map? I think it is not true, but I can't find any counter example!

Answer 1

Consider $f(\{(x,y): xy=1\}$. The im age is $\mathbb R \setminus \{0\}$.