The topological complexity of sets of convex differentiable functions (Q1128204)

If \(X\) is a Banach space let \(C(X)\) denote the set of all convex and continuous functions on \(X\) equipped with the topology of uniform convergence on bounded subsets of \(X\). The author proves the following theorem. Let \(X\) be a separable infinite-dimensional Banach space. Let \(M\) be a Polish space and \(A\) an analytic subset of \(M\). Then there is a continuous map \(\Gamma: M\to C(X)\) such that: (i) If \(t\in A\), then \(\Gamma(t)\) is not everywhere Gâteaux differentiable. (ii) If \(t\not\in A\), then the function \(\Gamma(t)\) is Fréchet-differentiable.