Let $B^n=\{x\in \mathbf{R}^n: |x|<1\}$$(n>2)$. Consider real analytic mappings $f_1:B^n\setminus \{0\}\to B^n$, $f_2:B^n\setminus \{0\}\to \mathbf{R}^n$ and $f_3: B^n\setminus \{0\}\to S^n$ satisfying the Jacobians $J_{f_i}$$(i=1,2,3)$ are nondegenerate everywhere. Is the isolated singularity $0$ always removable in the three cases , i.e. do there exist real analytic mappings $F_i(i=1,2,3)$ on $B^n$ such that $F_i=f_i$ on $B^n\setminus \{0\}$?

3empty? For $\det(J_{f_i})=0$ everywhere. $\endgroup$ – Pietro Majer Jul 6 at 7:14n-dimensional sphere, then OK sorry (and the above example essentially covers all three cases, 1,2 and 3). $\endgroup$ – Pietro Majer Jul 7 at 14:38