Given a function $f: X \rightarrow Y$ between metric spaces, we write $\Gamma_{f}$ for the subset $\{(x, f(x)) \mid x \in X\}$ of $X \times Y .$

(i) If $f$ is continuous, show that $\Gamma_{f}$ is closed in $X \times Y$.

(ii) If $Y$ is compact and $\Gamma_{f}$ is closed in $X \times Y$, show that $f$ is continuous.

(iii) Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\Gamma_{f}$ is closed but $f$ is not continuous.