For every such that , there exists a ball around and a ball around such that, for every in the ball around and every in the ball around , (this definition requires to be a metric space). If a preference relation is represented by a continuous utility function, then it is clearly continuous. By the theorems of Debreu (1954), the opposite is also true: Every continuous complete preference relation can be represented by a continuous ordinal utility function. Note that the lexicographic preferences are not continuous. For example, , but in every ball around (5,1) there are points with and these points are inferior to . This is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function. Uniqueness For every utility function v, there is a unique preference relation represented by v. However, the opposite is not true: a preference relation may be represented by many different utility functions. The same preferences could be expressed as any utility function that is a monotonically increasing transformation of v. E.g., if where is any monotonically increasing function, then the functions v and v give rise to identical indifference curve mappings. This equivalence is succinctly described in the following way: An ordinal utility function is unique up to increasing monotone transformation. In contrast, a cardinal utility function is unique up to increasing affine transformation. Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa. Monotonicity Suppose, from now on, that the set is the set of all non-negative real two-dimensional vectors. So an element of is a pair that represents the amounts consumed from two products, e.g., apples and bananas. Then under certain circumstances a preference relation is represented by a utility function . Suppose the preference relation is monotonically increasing, which means that "more is always better": Then, both partial derivatives, if they exist, of v are positive. In short: If a utility function represents a monotonically increasing preference relation, then the utility function is monotonically increasing. Marginal rate of substitution Suppose a person has a bundle and claims that he is indifferent between this bundle and the bundle . This means that he is willing to give units of x to get units of y. If this ratio is kept as , we say that is the marginal rate of substitution (MRS)