The findings on the ultimatum game are considered as belonging to the most robust experimental results. In this paper we present a slightly altered version of the mini ultimatum game of Bolton and Zwick (1995). Whereas in the latter exactly equal splits were feasible in our games these were replaced by nearly equal splits favoring (slightly) the proposer in one version and the responder in a second version. Such a minor change should not matter if behavior was robust. We found, however, a behavioral discontinuity in the sense that fair offers occur less often when equal splits are replaced by nearly equal splits. This has implications for theories incorporating fairness into economics.