  We find necessary and sufficient conditions for a Lipschitz map
$f:\mathbb{R}E\to X$, into a metric space to have the image with the
$k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting
feature of our approach is that despite the fact that we are dealing with
arbitrary metric spaces, we employ a variant of the classical implicit function
theorem. Applications include pure unrectifiability of the Heisenberg groups
and that of more general Carnot-Carath\'eodory spaces.
