Continuity of functions relative to their sets of discontinuity

It is well known that the set of points of continuity of a function f R→ R is a G_δ set, and that for every F_σ set D there is an f such that the set of points of discontinuities of f is D. In this paper we decompose D into the union of two disjoint F_σ sets U and V, such that f|V is discontinuous on V, and we show that V is maximal in the sense that there is an f such that f|D̄ is continuous on U.