  Consider a locally Lipschitz function $u$ on the closure of a possibly
unbounded open subset $\Omega$ of $\mathbb{R}^n$ with $C^{1,1}$ boundary.
Suppose $u$ is semiconcave on $\overline \Omega$ with a fractional
semiconcavity modulus. Is it possible to extend $u$ in a neighborhood of any
boundary point retaining the same semiconcavity modulus? We show that this is
indeed the case and we give two applications of this extension property. First,
we derive an approximation result for semiconcave functions on closed domains.
Then, we use the above extension property to study the propagation of
singularities of semiconcave functions at boundary points.
