= Solution
Now $G$ is continuous, so $G_x$ has no delta contribution. Integrate its normalized differential equation over $(\xi-\epsilon,\xi+\epsilon)$. The bounded lower-order terms have integrals tending to zero, while the integral of $G_{xx}$ is the difference of the two one-sided first <derivatives>. Hence the <jump condition> is
$$
\boxed{G_x(\xi^+;\xi)-G_x(\xi^-;\xi)=\frac1{\alpha(\xi)}.}
$$
The denominator is evaluated at the source point; it is not a variable $x$ left over after taking the limits. The positivity or negativity of this jump follows the sign convention used for the leading coefficient of $L$.
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