 Green function jump conditions

For $L=a_2(x)D^2+a_1(x)D+a_0(x)$ with $a_2(\xi )\ne 0$, a Green's function solving $L_{x}G=\delta (x-\xi )$ is homogeneous on either side of $\xi$, continuous there, and has first derivative jump $[G_x{]}_{-}^{+}=1/a_2(\xi )$. A nonzero jump in $G$ itself would produce an unwanted distributional derivative of the Dirac delta. Once continuity removes that term, integrating the equation across $\xi$ fixes the derivative jump. Endpoint boundary conditions complete the classical characterization.