If composition with a linear functional is continuous, is the function continuous?

Question

If $G$ is an open subset of $\mathbb{C}$ and $f:G \to X$ is a function such that for each $x^*$ in $X^*$, $x^*\circ f:G\to\mathbb{C}$ is analytic, then f is analytic. Will the statement still hold true if we replace both the instances of "analytic" with "continuous"?

Answer 1

Your notion of continuity with respect to linear functionals is called "weak continuity," and in general weak continuity doesn't imply continuity. Consider $\ell_2(\mathbb{C})$ and map $1/n \in\mathbb{C}\mapsto e_n$, $z\mapsto 0$ otherwise. This is weakly continuous at $0$, but not continuous there.