This chapter addresses the continuity property of demand relations with respect to prices. This property is regarded as an essential element in general equilibrium analysis of market transactions. Potential “discontinuity” of individual demand relations arises from the “discontinuity” of budget set relations. In order to identify the source of the discontinuity of these relations, the points in the consumption sets are first examined. Then, the set of these source points in each consumption set is analyzed. The set is defined as a subset of a consumption set X with respect to a price vector P, and is called the critical set of the consumption set. A related concept, that of a critical set in the wealth space, is also introduced.We then proceed to clarify the mathematical properties of these critical sets. Based on the concepts of critical sets, we examine the continuity properties of budget set relations and individual demand relations. In the final section of the chapter we bring up the issue of the survival of agents who do not possess sufficient wealth to participate in market transactions to obtain commodities in their consumption sets. This can result in empty demand sets. In order to address the issue of survival and the potential discontinuity of demand relations, the concept of subdemand is introduced.