The existence of large convex cones inside the family of real functions whose continuity set is a prescribed subset of a rather general topological space is established in this paper. This complements a number of recent assertions concerning lineability that have been obtained by several authors about functions defined on an interval of R. When the prescribed subset is a dense Gδ-set of a topological space satisfying appropriate conditions, an almost reciprocal of the Baire–Kuratowski theorem on 1-Baire functions is obtained, and again the family of functions satisfying the conclusion contains a large convex cone. For a dense Gδ-subset G of R, it is proved the existence of a dense vector subspace with maximal dimension in the space of continuous real functions all of whose nonzero members are derivable and the derivatives are continuous exactly at the points of G. Again, this complements a number of recent results made in the case that G is an open set of R, and yields as a consequence the existence of large vector spaces of 1-Baire functions with prescribed continuity set.