how
Suppose that V is a subset. By the continuity of f we have $\displaystyle f^{-1}(\overline{f(V)})$ is closed in X (f^-1 from Y to X maps closed/open sets to closed/open sets resp.).
Notice that $\displaystyle f(V) \subset \overline{f(V)} \implies V\subset f^{-1}(\overline{f(V)})$. As the closure is the smallest closed superset of V we have $\displaystyle \overline{V}\subset f^{-1}(\overline{f(V)})$. Now take f of both sides to get the result.
You don't need completeness (or the fact that it is a metric space).