1D Green function for a charged layer

sergiokapone

Problem Statement
Find 1D Green function for charged layer with distribution:

\phi=
\begin{cases}
0, x<-a \\
-\rho_0, -a<x<0 \\
\rho_0, 0<x<a \\
0, x>a
\end{cases}

With boundary condition $\phi(x \to -\infty) = 0$.
Relevant Equations
$\nabla^2\phi = -4\pi\rho$ and $\nabla^2G(x,x') = -4\pi\delta(x-x')$
I came across an example of a solution to finding the potential of a charged layer using the Green function (here, pdf). The standard algorithm for finding the Green function by boundary conditions for many problems is understandable:
\begin{align*}
G_\mathrm{Left} = Ax+ B \\
G_\mathrm{Right} = Cx + D \\
G_\mathrm{Right}' - G_\mathrm{Left}' = -4\pi
\end{align*}
since the Green function is linear in 1D problems, then using boundary conditions we find constants that are functions of primed coordinates:

But in this case I cannot understand how to find the Green function (how to determine constants) by the boundary condition $\phi(x \to -\infty) = 0$, which, as indicated in the link, is equal $G(x,x') = 4\pi x_{<}$. After all, the linear function does not disappear when striving to infinity. So, I need help.

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