Relationship between expected utility and independence axiom
If the preferences do not have an expected utility representation, then either the preferences are not continuous, or they do not satisfy the axiom  of independence.

For example in prospect theory, consumers display loss aversion, which  means that their preferences are not linear in probabilities. This is a case where preferences do not have the expected utility form, but the utility function is still continuous. This preferences violate the independence axiom.

Similarly, you could have preferences that do not admit the expected utility form, and satisfy the independence axiom, but not continuity.