Gauge transformations with Dirac point splitting are systematically discussed for the case of a pure Yang-Mills theory. These generalized gauge transformations are based on two ingredients: a fixed four-vector, which defines the point splitting, and a weight function, which gives an average over the amount of point splitting and which provides a cutoff in momentum space in the direction of the point splitting four-vector. From the requirement that the group property must be satisfied, it is found, starting from a simple ansatz, that an infinitesimal generalized gauge transformation takes the form of an infinite series in the coupling constant. Using induction on the order of the coupling constant, it is shown that all higher-order terms indeed exist and that they can be expressed in terms of the lower-order formulas. That there are such generalized gauge transformations suggests the possibility of a Yang-Mills field theory with mitigated divergences.