We can replace Continuity with the Archimedean property (or, 'You would accept some chance of a bad outcome from crossing the street.') By my reading, this ELU idea trivially follows Archimedes by ignoring the part of a compound 'lottery' that involves a sufficiently small probability. In which case it would violate Independence, and would do so by treating the two sides as effectively equal when the differing outcomes have small enough probability.