Abstract: The behavioral principle of continuity stipulates that preferences should be similar for similar random variables. We describe rich sets of random variables for which lower-semicontinuity with respect to the topology of convergence in measure becomes incompatible with convexity of strictly better sets while upper-semicontinuity becomes incompatible with convexity of strictly worse sets. Economic applications which involve complete preferences over rich sets must thus either give up risk aversion (loving, neutrality) or the behavioral principle of continuity whenever similarity of random variables is described by convergence in measure or by convergence in distribution. As one implication, any coherent risk measure defined on any standard vector space of random variables must be violating lower-semicontinuity.