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We study separate convexity for preferences and correspondences, and show that this weakening of the usual convexity postulate is strong enough to recover standard equivalences among continuity assumptions. For complete and transitive preferences, we establish equivalence theorems linking separate continuity, mixture continuity, Archimedean-type postulates, solvability and graph continuity, successively on product mixture sets and on Euclidean spaces. The results highlight the role of weaker axiomatic assumptions by yielding representations for multilinear cardinal utility, continuous separately quasiconcave ordinal utility in $n$-person decision problems, and a scalar Anscombe--Aumann setting. For non-ordered preferences, formulated as correspondences, we characterize the open graph property under separate convexity and weak section-continuity, generalizing results of Schmeidler, Shafer, and Bergstrom-Parks-Rader. Examples identify the boundaries of our results.
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