On the continuity of biconjugate convex functions (Q2781351)

\textit{S. Simons} [Set-Valued Analysis 7, No.~3, 255-294 (1999; Zbl 0992.47024)] constructed an example of a continuous convex function on~\(c_0\) whose Fenchel biconjugate \(f^{**}\) fails to be continuous as an extended real-valued function on~\(\ell^\infty\). The authors show that a Banach space~\(X\) is a Grothendieck space if and only if every continuous convex function on~\(X\) has a continuous biconjugate \(f^{**}\) on~\(X^{**}\), thus also answering a question raised by S.~Simons. Related characterizations and examples are given.