real analysis – Intermediate value property for Sobolev functions Answer

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real analysis – Intermediate value property for Sobolev functions

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

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

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

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