A consumer's preferences can be represented by a utility function if they satisfy properties P.1 through P.4, and one additional property called continuity. Continuity is probably the least intuitive property of preferences, yet it is not implausible.

Preferences are continuous if the set of all choices that are at least as good as a choice x' and the set of all choices that are no better than x' are both closed sets. In the notation of sets, this is written as {x : x x'} and {x : x' x} are both closed.

One definition of a closed set is that any sequence of points in the set that converges, converges to a point of the set. In this context, that means that for a sequence of points {x^{n}} with n = 1, 2, 3, ..., if x x^{n} for every x^{n} and if x^{n} converges to some consumption point x', then x x'.
Figure 6 shows an example of this. In the figure, if x x^{n} for every n, and if x^{n} converges to x', then continuity implies that x x'.

Figure 6: Sequence of points x^{n} that converge to x'.

If a consumer has a preference relation that
is complete, reflexive, transitive, strongly monotonic, and continuous, then these preferences can be represented by a continuous utility function u(x) such that
u(x) > u(x') if and only if x x'.

Proof : Let e = (1, 1, ..., 1). For each x, define u(x) by u(x) e x. Then u(x) is a utility function for the preferences if
1. such a function u(x) exists,
2. the function u(x) is unique, and
3. u(x) > u(x') if and only if x x'.

Let B = {a: a e x}. If x = (x_{1}, x_{2}), let y = (max{x_{1}, x_{2}}, max{x_{1}, x_{2}}). Then strong monotonicity implies that y x, so B is not empty.
Let W = {a: x a e}. Then 0 is an element of W, so W is not empty. By completeness, B W = {a : a 0}. Both B and W are closed sets, from property P.5 (continuity), so B W is not empty. Therefore, there is some a such that a e x. By strong monotonicity, if a' > a, then a' e x, and if a' < a, then x a' e, so a is unique. Let u(x) = a. So u(x) exists and it is unique.
Next, we want to show that u(x) represents the preferences . Suppose that x and y are two consumption levels and u(x) = a_{x} where a_{ x} e x. Let u(y) = a_{ y} where a_{ y} e y. If a_{ x} > a_{ y} then by monotonicty a_{ x} e a_{ y} e. By transitivity, x a_{ x} e a_{ y} e y. Finally, if x y, then a_{ x} e a_{ y} e so that a_{ x} > a_{ y}.
