Calculus of Variations

Abstract. With a map f : Ω → R,Ω ⊂ R, that belongs to the John Ball classAp,q(Ω) where n − 1 < p < n and q ≥ p/(p − 1) one can associate a set valued map F whose values F (x) ⊂ R are subsets ofR describing the topological character of the singularity of f at x ∈ Ω. Šverak conjectured that Hn−1(F (S)) = 0, where S is the set of points at which f is not continuous andHn−1 is the Hausdorff measure. The purpose of our paper is to confirm this expectation.