Summary: A convex (or concave) real-valued function, \(f\), on a dual Banach space \(P^\ast\) is continuous for the Mackey topology m\((P^\ast,P)\) if (and only if) it is Mackey continuous on bounded subsets of \(P^\ast\). Equivalence of Mackey continuity to sequential Mackey continuity follows when \(P\) is strongly weakly compactly generated, e.g., when \(P=L^1(T)\), where \(T\) is a set that carries a sigma-finite measure \(\sigma\). This result of Delbaen, Orihuela and Owari [F. Delbaen and K. Owari, Positivity 23, No. 5, 1051–1064 (2019; Zbl 1439.46023); F. Delbaen and J. Orihuela, J. Math. Anal. Appl. 485, No. 1, Article ID 123764, 15 p. (2020; Zbl 1433.91051)] extends their earlier work on the case that \(P^\ast\) is either \(L^\infty(T)\) or a dual Orlicz space. An earlier result of this kind is recalled also: it derives Mackey continuity from bounded Mackey continuity for a nondecreasing concave function, \(F\), that is defined and finite only on the nonnegative cone \(L^\infty_+\). Applied to a linear \(f\), the Delbaen-Orihuela-Owari result shows that the convex bounded Mackey topology is identical to the Mackey topology, i.e., cbm\((P^\ast,P)=\mathrm{m }(P^\ast,P)\); here, this is shown to follow also from Grothendieck’s Completeness Theorem. As for the bounded Mackey topology, bm\((P^\ast,P)\), it is conjectured here not to be a vector topology, or equivalently to be strictly stronger than m\((P^\ast,P)\), except when \(P\) is reflexive.