A $C^1$ function $f:\mathbb{R}^2\to \mathbb{R}$ whose critical value has a non-zero measure.

Question

Does there exist a function $f:\mathbb{R}^2\xrightarrow{C^1}\mathbb{R}$ such that the critical value has a non-zero measure?

It is not satisfying the condition of Sard's theorem, as in the Sard's theorem, we need at least $C^2$ regularity. So I believe such a function exists, but unable to construct.

Answer 1

Whitney constructed such examples, and it seems that this article is inspired by his work (I am unable to find Whitney's article "A function not constant on a connected set of critical points" free of access).