Surjective mappings whose differential is nowhere surjective

Abstract:

Examples of ${C^k}$-mappings $f:{\mathbb {R}^n} \to {\mathbb {R}^m},n \geq m > 2$, are given, with rank $df(x) \leq s$ at any $x \in {\mathbb {R}^n},2 \leq s < m$, but $f({\mathbb {R}^n}) = {\mathbb {R}^m}$, for any $k < (n - s + 2)/(m - s + 2)$. Thus a weak form of Sard’s theorem (if all the points in the source are critical, the image has measure zero) does not hold for mappings of low smoothness.