  We examine the one-dimensional quantum dynamics of a Schroedinger particle in
a potential represented by a generalized function of the form $U(x) = -\alpha
\delta (x) + \beta d(\delta (x))/dx$ superposed on a well behaved potential
$V(x)$. In this, we construct the full, exact Green's function for such a 1D
system analytically in closed form, taking account of a spatially variable mass
$m(x)$. Our result shows that there can be no electron transmission through the
$\beta \delta '(x)$- potential, regardless of the presence of the $V(x)$-
potential and $\alpha \delta (x)$, (with $\alpha \ne 0$).
