# Example of a rational and continuous preference relation that does not admit a utility representation

It is well-known that a rational, continuous and monotone preference relation $$\succeq$$ defined on $$\mathbb{R}^L$$ admits a utility representation.

I would like to understand why monotonicity is required. In other words, what is an example of a rational and continuous but non-monotone preference relation that does not have a utility representation?

• I don't think monotonicity is required for utility representation, at least in Debreu’s representation theorem. – Herr K. Jan 14 '19 at 20:05
• Thank you, you are right. I had never seen this result without the monotonicity axiom, which is probably used for simplifying the proof. I'll accept this answer if you post it. – Oliv Jan 14 '19 at 20:30

• (+1) Two related results may be worth noting. First, if $\succsim$ is defined on a finite or countable set, then rationality of $\succsim$ alone is both necessary and sufficient for utility representation. Second, if $\succsim$ is defined on an uncountable set, then rationality and continuity together are necessary and sufficient for (continuous) utility representation. – Herr K. Jan 15 '19 at 1:46