In this article, we prove a comparison result for viscosity solutions of a certain class of fully nonlinear, possibly degenerate, parabolic equations; the main new feature of this result is that it holds for any, possibly discontinuous, solutions without imposing any restrictions on their growth at infinity. The main application of this result which was also our main motivation to prove it, is the uniqueness of solutions to one-dimensional equations including the mean curvature equation for graphs without assuming any restriction on their behavior at infinity.