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In [[mathematics]], the '''inverse function theorem''' gives sufficient conditions for a vector-valued function to be [[invertible]] on an open region containing a point in its ___domain. The theorem can be generalized to maps defined on manifolds, and on infinite dimensional [[Banach space]]s.
The theorem states that if the [[total derivative]] of a function ''F'' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup> is invertible at a point ''p'' (i.e., the [[Jacobian determinant]] of ''F'' at ''p'' is nonzero), and ''F'' is [[continuously differentiable]] near ''p'', then it is an invertible function near ''p''. That is, an [[inverse function]] to ''F'' exists in some [[neighbourhood (mathematics)|neighborhood]] of ''F''(''p'').
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