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?
A necessary condition for a preference relation to be represented by a utility function is that the preference relation is rational (where a "utility function" is a real-valued function that assigns a higher or equal numerical value to bundle A than to bundle B, when A is weakly preferred to B).

Preferences are rational when they are complete (I can express my preferences for all conceivable bundles), and transitive (If I weakly prefer A to B and B to C, then I weakly prefer A to C).  

Given rationality, a sufficient condition is that the rational preference relation is continuous, i.e. if it is preserved under limits. 

Then, the related utility function is also continuous.  

Monotonicity is a property where we go from quantities to preferences, after assuming desirability ("goods" rather than "bads"). Local nonsatiation is actually the weaker assumption that is required for most of the theory.