Let $f: X \rightarrow Y$ be a continuous map between topological spaces. Let

$$G_{f}=\{(x, f(x)): x \in X\} .$$

(a) Show that if $Y$ is Hausdorff, then $G_{f}$ is closed in $X \times Y$.

(b) Show that if $X$ is compact, then $G_{f}$ is also compact.