Separable Metric Spaces: Ulam's Regularity Theorem

Let $X$ be a metric space that contains a finite dense subset $D$. Show that $X = D$.

Show that if two continuous mappings defined on a metric space $X$ take the same values on a dense subset, then they are equal.

Prove Lusin's Theorem for a finite Borel measure space ($X, \mathcal{B}(X), \mu$) where $X$ is a separable, complete metric space.