Problem sheet 2, question 2

Please note that the model answer shows that $NW(f)$ and $\overline{R(f)}$ are forward invariant, i.e. $f(NW(f))\subset NW(f)$ and $f(\overline{R(f)})\subset \overline{R(f)}$. I adjusted the questions accordingly. In fact, proving the opposite including $\supset$ is a little more involved and in the case of $NW(f)$ requires some additional condition (that $f$ is open).