Continuity of the Fenchel Transform of Convex Functions

Given a separated dual system $(E,E’)$, the Fenchel transform determines a pairing of the convex functions on E with the convex functions on $E’$. This operation is shown to have a continuity property. The result implies that the minimum set of a convex function varies in an upper-semicontinuous way with the function’s conjugate. Several convergence concepts for convex functions are discussed. It is shown for each of the two most useful that the Fenchel transform is not a homeomorphism.