Let $X,Y$ be complex projective varieties with $X$ irreducible, and let $f:X\dashrightarrow Y$ be a rational map. If $U\subseteq X$ is the largest open set where $f$ can be defined, is it true that $\mathrm{codim}_{X}(X\setminus U)\geq 2$. I know this is true if $X$ is smooth.

EDIT: In view of the inkspot's answer, I add: if $D\subset X$ is an irreducible divisor on $X$, with $D\cap\left(X\setminus\mathrm{sing}(X)\right)\neq\emptyset$, can it happen that $D\cap U=\emptyset$?

Thanks.