Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.

I know rationality over $\preccurlyeq$ does not imply a utility function. But rationality and continuity over $\preccurlyeq$ implies a utility function. But what about the reverse direction?

For example, $u(x) = sin(x) + 1$, is not rational, but is continuous, is it a utility function?

In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?

Given some irrational preferences, that can be represented by a function. If the function does not satisfy rationality (transitivity, completeness), does this imply it is not a utility function.

I know rationality over $\preccurlyeq$ does not imply a utility function. But rationality and continuity over $\preccurlyeq$ implies a utility function. But what about the reverse direction?

For example, $u(x) = sin(x) + 1$, is not rational, but is continuous, is it a utility function?

In my books I see a lot about the requirements needed to make a utility function, but given a function, what are the requirements for it to be a valid utility function?

My Answer A utility function is the representation of a preference relation $\preccurlyeq$. All preference relations are by assumption (or definition), rational. Given a function, if there does not exist any rational preference relation, then it must not be a utilty function.