It is well known that the dynamical system generated by Newton’s Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton’s Method of roots of a polynomial "going complex". More generally, we consider Newton’s Method for smooth real-valued functions of the formfμ(x)=g(x)+μ{f_\mu }(x) = g(x) + \mu,μ\mua parameter. Ifμ0{\mu _0}is a point of discontinuity of the mapμ→\mu \to(the number of roots offμ{f_\mu }), then, in the presence of certain nondegeneracy conditions, we show that there are values ofμ\munearμ0{\mu _0}for which the Newton function offμ{f_\mu }has nontrivial periodic attractors.