= Green function jump conditions
{c}
For $L=a_2(x)D^2+a_1(x)D+a_0(x)$ with $a_2(\xi)\ne0$, a <Green's function> solving $L_xG=\delta(x-\xi)$ 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.
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