I have a problem set stating that a competitive equilibrium does exist under a series of assumptions on the economy.
The question is "Show that the following six assumptions are needed for existence by constructing counterexamples to it where 5 of them are satisfied and one not".
The assumptions are continuity, convexity and local nonsatiation of preferences and closedness, convexity and inaction of the production set.
From what I got from the question and the TA explanations, it seems that if one of this properties fails, then there should be no competitive equilibrium. But I have actually found cases in which these properties are violated but an equilibrium exists. For example, in an Edgeworth box with bliss point preferences (which are not locally nonsatiated) we can find an equilibirum; the same happens when we have for example lexicographic (which are not continuous) and perfect substitutes.
SO my question is are these assumptions necessary for the existence of an equilibirum or not? And if not, what would be the purpose of the exercise?