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 $B^n\setminus \{0\}$?

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    $\begingroup$ Consider e.g. $x\mapsto x+ {\|x\|\over2} {\bf e_1} $, where $\|x\|$ is the Euclidean norm and ${\bf e_1}=(1,0,\dots,0)$. And isn't case 3 empty? For $\det(J_{f_i})=0$ everywhere. $\endgroup$ – Pietro Majer Jul 6 '19 at 7:14
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    $\begingroup$ Thank you so much. However, I think there exists a map satisfying case 3 based on your example. Consider the map $g:x\mapsto (x+\frac{||x||}{2}e_1,x_1^2+x_2^2+x_3^2)\in \mathbb{R}^4$. Then $G:x\mapsto \frac{g(x)}{||g(x)||}$ is in the case 3. $\endgroup$ – Sien Jul 7 '19 at 4:12
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    $\begingroup$ Oh I see, $S^n$ is the n-dimensional sphere, then OK sorry (and the above example essentially covers all three cases, 1,2 and 3). $\endgroup$ – Pietro Majer Jul 7 '19 at 14:38

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