Let $X, Y$ be topological spaces, and suppose $Y$ is Hausdorff.

(i) Let $f, g: X \rightarrow Y$ be two continuous maps. Show that the set

$$E(f, g):=\{x \in X \mid f(x)=g(x)\} \subset X$$

is a closed subset of $X$.

(ii) Let $W$ be a dense subset of $X$. Show that a continuous map $f: X \rightarrow Y$ is determined by its restriction $\left.f\right|_{W}$ to $W$.