Most convex functions are nice(*)

Abstract

Let : f is convex and lower semicontinuous}: this space with the metric of epiconvergence is compact, hence complete. For and , let fp be the function defined by fp(x)=f(x)-(p,x). Let be the set of real-valued functions which are differentiable on all of and s.t. fp is well-posed for every . Then, is a first category (i.e.: meager) set in .