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ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation D_xG(u) = t^2N-1D&sup2_xK(u) + t^N-1D_x,F(u) for x &#8712 R,t > 0 u(x,0) = A for x < 0, u(x,0) = B for x > 0 where A < B, and N > O are given constants; K(u) =^def &#8747^u_Ak(s)ds, G(u)=^def &#8747^u_Ag(s)ds, and F(u) =^def &#8747^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and F(B)G(B)G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity