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For $X = \mathbb{R}^2$, does there exist a quasi-concave and monotone function $f : X \to \mathbb{R}$ that has a level curve which is not path-connected?

Secondly, will every level curve necessarily intersect with the $x$-axis and the $y$-axis for the domain $X = \mathbb{R^2_{+}}$ ($0 \in \mathbb{R_{+}}$)?

If $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} > \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \implies f(x_1, x_2) > f(y_1, y_2)$ for all comparable $(x_1, x_2) \in X$ and $(y_1, y_2) \in X$, then we call $f$ (weakly) monotone.


Context : While doing Economics, I was wondering if convex and monotone preferences (assuming completeness and transitivity) can have indifference curves that are not path-connected. A more precise version of the question will be:

Given $X = \mathbb{R}^2$ and a preference relation $\succeq$ that is complete, transitive, convex and monotone, does there exist an indifference curve which is not path-connected?

Definitions:

  1. Convexity: $\forall x,y \in X: x \succeq y \implies \lambda x + (1-\lambda) y \succeq y \ \forall \lambda \in [0,1]$.
  2. Monotonicity: $\forall x, y \in X: x \geq y \implies x \succeq y$. [Here, if $x = (x_1, x_2)$ and $y = (y_1, y_2)$, then $x \geq y$ means $(x_1 \geq x_2) \land (y_1 \geq y_2)$.]
  3. Indifference curve (IC): Suppose $x \in X$. Then the set of elements that are indifferent to $x$ is given by $\text{IC}(x) := \{y \in X : y \succeq x \land x \succeq y \}$. If $y \in \text{IC}(x)$, we say $y \sim x$ where $\sim$ denotes indifference.
  4. Completeness and transitivity is defined similar to any other binary operator.

This is a more precise version of the problem as there may not (necessarily) exist a function $f$ that describes $\succeq$. But since this is MSE and I did not want to burden with definitions, I decided to assume the existence of $f$.

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