  Wiener process with instantaneous reflection in narrow tubes of width
{\epsilon}<<1 around axis x is considered in this paper. The tube is assumed to
be (asymptotically) non-smooth in the following sense. Let $V^{\epsilon}(x)$ be
the volume of the cross-section of the tube. We assume that
$V^{\epsilon}(x)/{\epsilon}$ converges in an appropriate sense to a non-smooth
function as {\epsilon}->0. This limiting function can be composed by smooth
functions, step functions and also the Dirac delta distribution. Under this
assumption we prove that the x-component of the Wiener process converges weakly
to a Markov process that behaves like a standard diffusion process away from
the points of discontinuity and has to satisfy certain gluing conditions at the
points of discontinuity.
