  We show that local deformations, near closed subsets, of solutions to open
partial differential relations can be extended to global deformations, provided
all but the highest derivatives stay constant along the subset. The
applicability of this general result is illustrated by a number of examples,
dealing with convex embeddings of hypersurfaces, differential forms, and lapse
functions in Lorentzian geometry.
  The main application is a general approximation result by sections which have
very restrictive local properties an open dense subsets. This shows, for
instance, that given any $K \in \mathbb{R}$ every manifold of dimension at
least two carries a complete $C^{1,1}$-metric which, on a dense open subset, is
smooth with constant sectional curvature $K$. Of course this is impossible for
$C^2$-metrics in general.
