This paper is concerned with instantaneous jumps in the state trajectory of a singular system. It is shown that such jumps generically exist at the initial time due to inconsistent initial states. The formula for the jumps is derived, which indicates that the jumps linearly depend on the initial state. Moreover, the set of all initial states which do not cause any state jumps for any output feedback is characterized and is revealed to be a linear subspace. In the meantime, the necessary and sufficient conditions are given for the existence of an output feedback to suppress completely the jumps for a given initial state. Finally, we formulate the problem of designing an output feedback not only to stabilize the plant but also to achieve minimum initial jumps as a minimization problem.