An implicit function theorem

Abstract

Suppose thatF:D⊂R ⁿ×R^m→Rⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ∂₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.