If f is a continuous mapping of a metric space $\displaystyle X$ into a metric space $\displaystyle Y$, prove that $\displaystyle f(\overline{E}) \subset \overline{f(E)}$ for every set $\displaystyle E \subset X$. Show by an example that $\displaystyle f(\overline{E})$ can be a proper subset of $\displaystyle \overline{f(E)}$.