In this paper the author finds a continuously differentiable function \(f: [-1,1]^ 2\to [-1,1]\) such that for any \(u\in [-1,1]\) the set \(f^{- 1}(u)\) cannot be covered by a countable number of rectifiable curves. This result gives the negative answer to the question of D. Preiss whether for any Lipschitz function \(f: \mathbb{R}^ 2\to\mathbb{R}\) and for almost all real numbers \(u\) the level set \(f^{-1}(u)\) can be covered by a countable number of rectifiable curves. Let us remember that for almost all \(u\) this is true up to a remainder of one-dimensional Hausdorff measure zero.