  Discontinuous initial wave functions or wave functions with discontintuous
derivative and with bounded support arise in a natural way in various
situations in physics, in particular in measurement theory. The propagation of
such initial wave functions is not well described by the Schr\"odinger current
which vanishes on the boundary of the support of the wave function. This
propagation gives rise to a uni-directional current at the boundary of the
support. We use path integrals to define current and uni-directional current
and give a direct derivation of the expression for current from the path
integral formulation for both diffusion and quantum mechanics. Furthermore, we
give an explicit asymptotic expression for the short time propagation of
initial wave function with compact support for both the cases of discontinuous
derivative and discontinuous wave function. We show that in the former case the
probability propagated across the boundary of the support in time $\Delta t$ is
$O(\Delta t^{3/2})$ and the initial uni-directional current is $O(\Delta
t^{1/2})$. This recovers the Zeno effect for continuous detection of a particle
in a given domain. For the latter case the probability propagated across the
boundary of the support in time $\Delta t$ is $O(\Delta t^{1/2})$ and the
initial uni-directional current is $O(\Delta t^{-1/2})$. This is an anti-Zeno
effect. However, the probability propagated across a point located at a finite
distance from the boundary of the support is $O(\Delta t)$. This gives a decay
law.
