= Solution
Suppose the piecewise smooth <Green function> has jump $[G]_\xi=G(\xi^+;\xi)-G(\xi^-;\xi)$. Its first distributional <derivative> then contains $[G]_\xi\delta(x-\xi)$, and its second contains $[G]_\xi\delta'(x-\xi)$. In the normalized equation from part (a), the first-derivative term can supply a delta but no delta <derivative>. The right side has only a delta. Thus its delta-derivative coefficient must vanish:
$$
\boxed{[G]_\xi=0.}
$$
Equivalently, the two homogeneous branches must join continuously. This is the continuity part of the <jump condition> for a second-order <Green function>.
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