Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function.

Question: For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\mu(f^{-1} ([a, b]) > 0$?

Note: Here $\mu$ denotes the Lebesgue measure.

Let $d \geq 2$, and let $f \in W^{1, 1} (\mathbb R^d)$ be a Sobolev function.

Question: For any $a, b \in \mathbb R$ such that $\text{essinf } f \leq a < b \leq \text{esssup } f$, is it true that $\mu\left(f^{-1} ([a, b])\right) > 0$?

Note: Here $\mu$ denotes the Lebesgue measure.