This article treats discontinuous fold bifurcations of periodic solutions of discontinuous system. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle causing a discontinuous bifurcation. Numerical examples are treated which show discontinuous fold bifurcations. The discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions.