Continuous functions

[Grigorios Kostakos]

Let $(X,\rho)$, $(Y,d)$ two metric spaces and $f,g:X\longrightarrow Y$ two continuous functions.

1. Prove that the set $F=\big\{x\in X\;|\; f(x)=g(x) \big\}$ is closed set of $X$.
2. If $D$ is a dense subset of $X$, such that $f (x) = g(x)$, for every $x\in D$, prove that $f = g$.