  Functions with discontinuities appear in many applications such as image
reconstruction, signal processing, optimal control problems, interface
problems, engineering applications and so on. Accurate approximation and
interpolation of these functions are therefore of great importance. In this
paper, we design a moving least-squares approach for scattered data
approximation that incorporates the discontinuities in the weight functions.
The idea is to control the influence of the data sites on the approximant, not
only with regards to their distance from the evaluation point, but also with
respect to the discontinuity of the underlying function. We also provide an
error estimate on a suitable {\it piecewise} Sobolev Space. The numerical
experiments are in compliance with the convergence rate derived theoretically.
