We show that the classical geometric forms of the axiom of Hahn-Banach are equivalent to the continuous Hahn-Banach property. It follows that Gâteaux differentiable uniformly convex Banach spaces satisfy the continuous Hahn-Banach property. We also prove that the axiom of dependent choices, which is equivalent to Ekeland's variational principle, implies that Gâteaux differentiable Banach spaces satisfy the continuous Hahn-Banach property.