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We investigate the value function of the Bolza problem of theCalculus of Variations
$$ V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds +\varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\},$$ 
with a lower semicontinuous Lagrangian L and a final cost $ \varphi $ 
,andshow that it is locally Lipschitz for t>0whenever L is locally bounded. It also satisfiesHamilton-Jacobi inequalities in a generalized sense.When the Lagrangian is continuous, then the value function is theunique lower semicontinuous solutionto the corresponding Hamilton-Jacobi equation, while for discontinuousLagrangian we characterize the value function by using the socalled contingent inequalities.
