Range of continuous transformation on closes set

Question

let $f$ be a continuous transformation and $F$ closed set. Prove that the range $f(F)$ does not have to be closed.

Answer 1

For example $f(x)=\operatorname{atan}x$ and $f(\mathbb{R})$ is an open interval.

Answer 2

Take a projection $\Bbb R\times\Bbb R\to\Bbb R$ which is given by $p(x,y)=x$. So, if you choose the closed set $F=\{(x,y)\mid xy=1\}$ then $p(F)$ is not closed.