Is the biconjugate of a continuous functions also continuous? Let $f:\mathbb{R}^n\to\mathbb{R}$ be given and assume that $|f(x)|\leq C|x|^2$. Is it true that the bi-(convex/Fenchel)-conjugate $f^{**}$ is also continuous. It was claimed in a book without a proof. Of course I checked the related Fenchel relations but these only provide a semicontinuity result for the (bi)conjugate.

$f^{**}\le f$ and $f$ is bounded above on an open set makes $f^{**}$ bounded above on an open set so because $f^{**}$ is convex lower semicontinuous it is continuous on the interior of its domain.