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Prove that if the functions $f_1,...f_n$ in the statement of the implicit function theorem are assumed to be $k$ times differentiable (i.e., all partial derivatives of order k exist and are continuous), then the same is true of all the component functions $\varphi_1,...,\varphi_n$ of $\varphi.$

I know I should differentiate $f(x,\varphi(x))=0$ and express the Jacobi matrix of $\varphi$ in terms of the Jacobi matrix of $f$ and then explain why the Jacobi matrix of $\varphi$ is continuously differentiable.

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  • $\begingroup$ I know I should differentiate f(x,p(x))=0 and express the Jacobi matrix of p in terms of the Jacobi matrix of f and then explain why the Jacobi matrix of p is continuously differentiable. $\endgroup$ – user142691 Apr 12 '14 at 21:01
  • $\begingroup$ p should be phi in the above comment... $\endgroup$ – user142691 Apr 14 '14 at 19:45

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