It is said that an absolute continuous function is differentiable almost everywhere; a locally Lipschitz function is absolute continuous and therefore differentiable almost everywhere.
I am confused about the following example: $$V(x_1,x_2)=|x_1|+|x_2|$$ It is easy to check $V$ is locally Lipschitz. However, $V$ is not differentiable at $x_1=0$ or $x_2=0$. The two (infinite-length lines) sets $x_1=0$ and $x_2=0$ are not of measure zero, right? Then does it conflict with that $V$ is differentiable almost everywhere? Where am I mistaken?