# Can quasi-concave and monotone functions level curves that are not path-connected?

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 $$f$$.