If all compositions of a vector-valued function with its linear functionals are analytic, is the function itself analytic?

Question

This is taken from Conway's A course in functional Analysis (p. 198, Exercise 4):

Let $\mathscr{X}$ be a Banach space and $G \subset \mathbb{C}$ open. If $f: G \rightarrow \mathscr{X}$ is such that for each $x^{*} \in \mathscr{X}^{*}$, $(x^{*} \circ f): G \rightarrow \mathbb{C}$ is analytic, prove $f$ is analytic itself. If the word "continuous" is substituted for both occurrences of the word "analytic", is the preceding statement still true?