Problem: Let $(X,d_X)$ and $(Y, d_Y)$ be metric spaces. Suppose that $d_X$ is the discrete metric. Show that any function $f: (X, d_X) \to (Y, d_Y)$ is continuous.
Solution: We will show inverse image of any open set in $Y$ must be open in $X$. Note that $d_X$ is discrete metric, so every set is open in $X$. If $U$ is open in $Y$, then $f^{-1} (U) \subseteq X$ must be open in $X$. This proves that $f$ is continuous.