  We study the possibility of supersymmetry (SUSY) in quantum mechanics in one
dimension under the presence of a point singularity. The system considered is
the free particle on a line R or on the interval [-l, l] where the point
singularity lies at x = 0. In one dimension, the singularity is known to admit
a U(2) family of different connection conditions which include as a special
case the familiar one that arises under the Dirac delta-potential. Similarly,
each of the walls at x = l and x = -l admits a U(1) family of boundary
conditions including the Dirichlet and the Neumann boundary conditions. Under
these general connection/boundary conditions, the system is shown to possess an
N = 1 or N = 2 SUSY for various choices of the singularity and the walls, and
the SUSY is found to be `good' or `broken' depending on the choices made. We
use the supercharge which allows for a constant shift in the energy, and argue
that if the system is supersymmetric then the supercharge is self-adjoint on
states that respect the connection/boundary conditions specified by the
singularity.
