The further a curve is from the origin, the greater is the level of utility. The slope of the curve (the negative of the marginal rate of substitution of X for Y) at any point shows the rate at which the individual is willing to trade off good X against good Y maintaining the same level of utility. The curve is convex to the origin as shown assuming the consumer has a diminishing marginal rate of substitution. It can be shown that consumer analysis with indifference curves (an ordinal approach) gives the same results as that based on cardinal utility theory — i.e., consumers will consume at the point where the marginal rate of substitution between any two goods equals the ratio of the prices of those goods (the equi-marginal principle). Revealed preference Revealed preference theory addresses the problem of how to observe ordinal preference relations in the real world. The challenge of revealed preference theory lies in part in determining what goods bundles were foregone, on the basis of them being less liked, when individuals are observed choosing particular bundles of goods. Necessary conditions for existence of ordinal utility function Some conditions on are necessary to guarantee the existence of a representing function: Transitivity: if and then . Completeness: for all bundles : either or or both. Completeness also implies reflexivity: for every : . When these conditions are met and the set is finite, it is easy to create a function which represents by just assigning an appropriate number to each element of , as exemplified in the opening paragraph. The same is true when X is countably infinite. Moreover, it is possible to inductively construct a representing utility function whose values are in the range . When is infinite, these conditions are insufficient. For example, lexicographic preferences are transitive and complete, but they cannot be represented by any utility function. The additional condition required is continuity. Continuity A preference relation is called continuous if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions: For every , the set is topologically closed in with the product topology (this definition requires to be a topological space). For every sequence , if for all i and and , then .