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Let $X$ be a variety and $H$ a Cartier divisor on $X$. We prove that if $H$has Du Bois (or DB) singularities, then $X$ has Du Bois singularities near $H$.As a consequence, if $X \to S$ is a family over a smooth curve $S$ whosespecial fiber has Du Bois singularities, then the nearby fibers also have DuBois singularities. We prove this by obtaining an injectivity theorem forcertain maps of canonical modules. As a consequence, we also obtain arestriction theorem for certain non-lc ideals.

Du Bois singularities deform