  We investigate the value function of the Bolza problem of the Calculus of
Variations $$
  V (t,x)=\inf \{\int_{0}^{t} L (y(s),y'(s))ds + \phi(y(t)) : y \in W^{1,1}
(0,t; R^n) ; y(0)=x \}, $$ with a lower semicontinuous Lagrangian $L$ and a
final cost $\phi$, and show that it is locally Lipschitz for $t>0$ whenever $L$
is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a
generalized sense.
  When the Lagrangian is continuous, then the value function is the unique
lower semicontinuous solution to the corresponding Hamilton-Jacobi equation,
while for discontinuous Lagrangian we characterize the value function by using
the so called contingent inequalities.
