Here is the problem:

Let $f : X \to Y$ be a continuous map between topological spaces. Let $E$ be a dense subset of $X$ (that is, $\operatorname{Cl} E = X$), where $\operatorname{Cl}E$ represents the closure of $E$. Prove: $\operatorname{Cl} f(E) = f(X)$

I know continuity of $f$ implies $f(\operatorname{Cl} E) \subseteq \operatorname{Cl} f (E)$, but I have no idea to show the converse.