Let $f\colon \mathbb R\to\mathbb R$ and let $D_f$ denote the set of points of discontinuity of $f$. First it is proved that if $f|_{D_f}$ is continuous, then $D$ is a nowhere dense, $F_\sigma$ set. The major result is that if $D$ is a nowhere dense, $F_\sigma$ set then there is a function $f$ such that $D_f=D$ and $f|_D$ is continuous. Finally it is shown that such functions are of Baire class one.