This paper gives a unified and simple treatment of three related questions in the demand theory of the weak axiom: (i) Is there an elementary, i.e., non-fixed point theoretic, proof of equilibrium existence when the excess demand function of an economy satisfies the weak axiom? (ii) What conditions are sufficient for a non-transitive preference to generate a continuous demand function? Note that such a demand must satisfy the weak, though not necessarily the strong, axiom. This motivates the next question. (iii) Given a function that satisfies the weak axiom, can we find a (non necessarily transitive) preference that generates it? To answer this first question, we give a proof using the separating hyperplane theorem. With the help of this result, we identify a class of non-transitive preferences which generate continuous demand functions, and within which any demand function satisfying the weak axiom can be rationalized.
