Weak continuity implies continuity

Question

This problem is from $\textit{Conway's A course in Functional analysis 2ed, pge 198, problem 4}$. The problem is: If $\mathscr{G}$ is an open subset $\mathbb{C}$, $\mathscr{B}$ is a Banach space and $$f:\mathscr{G}\rightarrow \mathscr{B}$$ is a map such that for each $\mathscr{x^*}\in \mathscr{B^*}$ the fucntion $\mathscr{x^*\circ} \ \textit{f}$ is continuous. Show that $\textit{f}$ is also continuous.