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If $F = (f_1, f_2):R^3→R^2$is a continuously differentiable function on $R^3$ and $a \in R^3$ is a point at which $DF (a)$ has rank 2, then there exists a continuously differentiable function $f_3:R^3 →R$ such that the function $ \hat{F} = (f_1 , f_2 , f_3 ) : R^3 → R^3$ has an inverse function which is continuously differentiable in an open neighbourhood of $a$.
I could'nt find any way to show $\hat{F}$ has such an inverse.
Let $\hat{F}(\bar{x})=(f_1(\bar{x}),f_2(\bar{x}),z)$ for any $\bar{x}=(x,y,z) \in R^3$. Since partial Jacobian of $F$ is not equal to zero at $a \in R^3$, determinant of $D\hat{F}$ at $a$ is also non-zero. Moreover both $f_1$ and $f_2$ are continuously differentiable, which implies that $\hat{F}$ is also continuously differentiable on $R^3$, thus we can use inverse function theorem (in multidimensional Euclidean spaces) to get an open subset $W \subset R^3$, $a\in W$, such that $\hat{F}$ is injective on $W$, and $\hat{F}^{-1}$ is continuously differentiable on $\hat{F}(W)$. We are done.
