[Math] How to show that continuous functions between metric spaces agree on a closed set

Question

Let $(X,d)$ and $(Y,d')$ be metric spaces, and let $D$ be a dense subset of $X$. Show that:

If $f:X\to Y$ and $g:X\to Y$ be continuous, then the set $\{x\in X\mid f(x)=g(x)\}$ is closed.

Answer 1

Where does the set $D$ come into picture? This is an irrelevant piece of information needed to prove the question. Below is a hint.

HINT

Using the fact that continuity implies sequential continuity, try to prove that the set $\{x \in X \vert f(x) = g(x)\}$ contain its limit points.