If the Jacobian of a differentiable function is singular at a zero of the function, any piecewise linear approximation to it may not have a zero, even when the function has one. In this paper we present a technique for perturbing such functions so that the recent fixed point algorithms that trace zeros of piecewise linear homotopies will succeed in finding a zero of such functions. We also show how to unperturb in case the Jacobian is nonsingular at the solution, and thus not impede the super linear convergence attained by these algorithms.