  In this paper, we study the holonomic $D$-modules when $D$ is the ring of
$k$-linear differential operators on $A = k[\Gamma]$, the coordinate ring of an
affine monomial curve over the complex numbers $k = \mathbb C$. In particular,
we consider the graded case, and classify the simple graded $D$-modules and
compute their extensions. The classification over the first Weyl algebra $D =
A_1(k)$ is obtained as a special case.
