\(C^{\infty}\) maps may increase \(C^{\infty}\)-dimension (Q1096218)

The \(C^{\infty}\)-dimension of an arbitrary subset in a smooth manifold is defined to be the least integer m such that the subset is contained in a countable union of \(C^{\infty}\)-submanifolds of dimension m in the given manifold [see \textit{M. Gromov}, Partial differential relations (Springer 1986), Section (1.3.2)]. Gromov observed that the Thom equisingularity theorem shows that the \(C^{\infty}\)-dimension is monotone nonincreasing under generic \(C^{\infty}\) maps. In this paper the author provides an example and shows that there is a \(C^{\infty}\) function \(\chi\) : \(R\to R^ 2\) such that im \(\chi\) cannot be covered by a countable union of nonsingular \(C^ 1\) curves.