An Application: Runge's Theorem

Let $\mu$ be a compactly supported measure on $\mathbb {C}$ that is boundedly absolutely continuous with respect to area measure. Show that $\hat{\mu }$ is continuous on $\mathbb {C}_\infty$.

Let $m =$ Lebesgue measure on $[0,1]$. Show that $\hat{m}$ is not continuous at any point of $[0,1]$.