ABSTRACT
This chapter argues that if one rejects continuity in so-called 'extreme' cases, as many think one should, then there are deep and irresolvable difficulties for expected utility theory. To the contrary, if one rejects the principle of continuity in certain 'extreme' cases, then one must reject continuity even in the 'easy' cases for which it seems most plausible, reject the axiom of transitivity, or reject the principle of substitution of equivalence. Expected utility theory is a powerful and widely influential theory. It underlies game theory, decision theory, and much of modern economics. Continuity is important for expected utility theory. In response to the preceding argument, some people defend the principle of continuity even in so-called 'extreme' cases of the sort Vallentyne imagines. Correspondingly, there remains reason to question the axiom of transitivity, the principle of substitution of equivalence, or continuity even in so-called 'easy' cases.
