# The Continuity Axiom

This lecture explains the continuity axiom of expected utility theory.

Takeaway Points

1. Suppose you prefer A to B to C. The continuity axiom says that a unique probability p exists such that you are indifferent between a lottery of A with probability p and C with probability 1 – p and receiving B with certainty.
2. Continuity disallows a discontinuous jump in preferences over the lotteries, hence the name.
3. It also rules out lexicographic preferences across multiple continuous dimensions. For example, suppose a car can have a continuous amount of horsepower and a continuous range of colors between white and red. Suppose someone is unwilling to sacrifice any amount of horsepower to change the color. However, holding horsepower fixed, the individual prefers more red to less. Expected utility cannot represent these preferences.

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