A relation is locally the graph of a function wherever the derivative block in the solved-for variables is invertible.Suggest a slogan
Suppose \(f\in C^1(\RR^{m+n}, \RR^n)\), written \(f(x,y)\) with \(x\in\RR^m\) and \(y\in\RR^n\), that \(f(a, b) = 0\), and that the partial derivative in the second block, \(\partial f/\partial y\, (a,b)\), is an invertible \(n\times n\) matrix. Then there exists a neighborhood \(U\subseteq \RR^m\) containing \(a\) and a unique \(g\in C^1(U, \RR^n)\) such that \(g(a) = b\) and \(f(x, g(x)) = 0\) for all \(x\in U\).
A relation is locally the graph of a function wherever the block of the derivative belonging to the solved-for variables is nonsingular.
See [@Mun91, Section 9, Theorem 9.2].
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