$\newcommand\R{\mathbb R}$This is not true in general. E.g., let $p(s):=(\arctan s,0,\dots,0)$. Then $$\bigcup_{s\in\R}N_s=(-\pi/2,\pi/2)\times\R^{n-1}\ne\R^n.$$
Iosif Pinelis
- 127.7k
- 8
- 107
- 229