Show that there is no continuous utility function which represents the lexicographic preferences Show that there is no continuous utility function which represents the
lexicographic preferences $\mathscr{L}$ given by $(x_1, x_2) \succeq  (y_1, y_2)$ if and only if $x_1 &gt; y_1$ or $x_1 = y_1$ and $x_2 &gt; y_2$.
Hint:

To ensure the existence of utility function representation, we must have continuous preferences (and this is a sufficient condition). From the Mas-Colell book:

The preference relation $\succeq$ on $X$ is continuous if it is preserved under limits. That is, for any sequence of pairs ${(x^n, y^n)}^\infty_{n=1}$ with $x^n \succeq y^n$ for all $n$, $x = \lim_{n \to \infty} x^n$ and $y = \lim_{n \to \infty} y^n$, we have $x \succeq y$.

We then look at the definition of utility function representation.

The function $U$ represents $\succeq$ on $X$ if $\forall \ x,y \in X, x \succeq y \iff u(x) \geq u(y)$, where $u: X \to \mathbb{R}$.

Though it may be easier just to show that the preferences themselves are not continuous, and thus there is no utility function representation, continuous or not.