  Safety and liveness are elementary concepts of computation, and the
foundation of many verification paradigms. The safety-liveness classification
of boolean properties characterizes whether a given property can be falsified
by observing a finite prefix of an infinite computation trace (always for
safety, never for liveness). In quantitative specification and verification,
properties assign not truth values, but quantitative values to infinite traces
(e.g., a cost, or the distance to a boolean property). We introduce
quantitative safety and liveness, and we prove that our definitions induce
conservative quantitative generalizations of both (1)~the safety-progress
hierarchy of boolean properties and (2)~the safety-liveness decomposition of
boolean properties. In particular, we show that every quantitative property can
be written as the pointwise minimum of a quantitative safety property and a
quantitative liveness property. Consequently, like boolean properties, also
quantitative properties can be $\min$-decomposed into safety and liveness
parts, or alternatively, $\max$-decomposed into co-safety and co-liveness
parts. Moreover, quantitative properties can be approximated naturally. We
prove that every quantitative property that has both safe and co-safe
approximations can be monitored arbitrarily precisely by a monitor that uses
only a finite number of states.
