Abstract: When preferences are expressed in natural language, it is not uncommon to encounter discontinuities. For example, "We will never risk people's lives" seems to suggest a discontinuity at zero probability of an unacceptable outcome (death). Should such statements be taken at face value? We take an axiomatic approach, asking what modes of behavior are and are not compatible with such statements. Specifically, for decision under risk, we weaken the vMN axioms to derive two rather natural versions of discontinuity "at zero".