Inversion of adjunction for rational and Du Bois pairs

We prove several results about the behavior of Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the nonpair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the nonpair setting this asserts that if a family over a smooth base has a special fiber X-0 with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near X-0.