# Can every continuous preference relation be represented by a discontinuous utility function?

In this question, it is shown that a continuous preference relation can have a discontinuous utility function.

Is it true in general that every continuous preference relation must have a discontinuous utility function?

I know that a continuous preference relation must have a continuous utility representation, and I am trying to compose such a continuous function with an increasing but discontinuous function (which will be a utility function for the preference relation), but since the composition of a continuous and discontinuous function can be continuous, it is not trivial to show this composition to be discontinuous.

That is not true. Consider the continuous preference relation on $$\mathbb{R}^n_+$$ represented by utility function $$u(x)= 0$$. This does not have a discontinuous utility representation. In fact, this is the only continuous preference relation on $$\mathbb{R}^n_+$$ that does not have a discontinuous utility representation.
• Would this mean that $\succsim=\mathbb R_+^n\times\mathbb R_+^n$ so that there is perfect indifference: $x\sim y$ for all $x,y\in\mathbb R_+^n?$