Let $f:(X, d) \rightarrow(Y, e)$ be a function between metric spaces.

(a) Give the $\epsilon-\delta$ definition for $f$ to be continuous. Show that $f$ is continuous if and only if $f^{-1}(U)$ is an open subset of $X$ for each open subset $U$ of $Y$.

(b) Give an example of $f$ such that $f$ is not continuous but $f(V)$ is an open subset of $Y$ for every open subset $V$ of $X$.