Green's Function and Discontinuous Source

Question

I have recently been studying Green's functions; however, I have only been exposed to continuous forcing terms. I was wondering about the equation \begin{equation*} \begin{split} &A y''(x)+By(x)=f(x) \\ &y'(0)=y'(b)=0 \end{split} \end{equation*} where $f$ is a discontinuous function and $b > 0$. If I assume $f$ has a finite number of discontinuities, it seems possible for the question to lack uniqueness. Is there any way to get all of the solutions in terms of the Green's function (provided $A,B,b$ are chosen so that the homogeneous problem is well-posed)? Also when the homogeneous problem is well-posed, does that imply a solution will always exist for a discontinuous $f$? Thanks in advance. A reference on discontinuous forcing terms and Green's functions would be greatly appreciated if known.