Implicit function theore

یا نور

Let f : R^n+m → R^m be a continuously differentiable function, and let R^n+m have coordinates (x, y). Fix a point (a₁,...,a_n,b₁,...,b_m) = (a,b) with f(a,b)=c, where c∈ R^m. If the matrix [(∂f_i/∂y_j)(a,b)] is invertible, then there exists an open set U containing a, an open set V containing b, and a unique continuously differentiable function g:U → V such that

Regularity

It can be proven that whenever we have the additional hypothesis that f is continuously differentiable up to k times inside U×V, then the same holds true for the explicit function g inside Uand

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