The degrees-of-freedom of a K-user Gaussian interference channel (GIC) has been defined to be the multiple of (1/2)log ₂ P at which the maximum sum of achievable rates grows with increasing power P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GICs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are nonzero rational numbers. More specifically, for all K > 2, we find a class of K-user GICs that is dense in the GIC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIC with nonzero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.