Let $X$ be a (complex) banach space, $U$ be an open subset of $\mathbb{C}$ and $f: U \to X$ be a function that is completely arbitrary except that it satisfies the property that for any continuous linear functional $l$ on $X$, $l \circ f$ is complex analytic in the usual sense. Is it possible to deduce from this that $f$ is continuous? What about strongly analytic? (This means that the usual limit of the difference quotient exists in the norm of $X$.)
Can strong analyticity be concluded if I assume the weak analyticity condition plus continuity?