Let $M$ be an n-dimensional Riemannian manifold, and $f: M\to S^n$ be a smooth map, $df$ be the differential of $f$. Set
$$U=\{x\in \text{supp}\ (df)| |df|<1/2\}$$
$$V=\{x| |\Lambda^2(df)|>1/2\}$$
where $\Lambda^2(df)$ is the induced map of $df$ on the exterior product $\Lambda^2(TM)$.
I'm a little confused with the conclusion: $$\overline{V}\cap \overline{U}=\emptyset.$$
Why are they intersect in the case that inequality becomes equality at $1/2$?
Could you give me some help with the details? Thanks!