= Solution
The solution $x+\lambda$ satisfies the left <boundary condition>, while $e^{-x}$ satisfies decay at infinity. The proposed expression for $x<\xi$ is therefore the correct left homogeneous solution. For $x>\xi$, write $G=Ce^{-x}$. Continuity at $x=\xi$ gives
$$
Ce^{-\xi}=-\frac{\xi+\lambda}{\xi+\lambda+1}.
$$
Hence the <Green function> is
$$
\boxed{
G(x;\xi)=
\begin{cases}
-\dfrac{x+\lambda}{\xi+\lambda+1},&0\leq x<\xi,\\[6pt]
-\dfrac{\xi+\lambda}{\xi+\lambda+1}e^{\xi-x},&x>\xi.
\end{cases}}
$$
Indeed, its derivative has the required unit jump,
$$
G_x(\xi^+;\xi)-G_x(\xi^-;\xi)=1,
$$
which produces the <Dirac delta function>[Dirac delta distribution] in $L[G]$.
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