  We establish a unique continuation property for solutions of the differential
inequality $|\nabla u|\leq V|u|$, where $V$ is locally $L^n$ integrable on a
domain in $\mathbb R^n$. A stronger uniqueness result is obtained if in
addition the solutions are locally Lipschitz. One application is a finite order
vanishing property in the $L^2$ sense for the exponential of $W^{1,n}$
functions. We further discuss related results for the Cauchy-Riemann operator
$\bar\partial$ and characterize the vanishing order for smooth extension of
holomorphic functions across the boundary.
