An implicit function theorem

Abstract

Suppose that F:D⊂Rn×Rm→Rn, with F(x0, y0)=0. The classical implicit function theorem requires that F is differentiable with respect to x and moreover that ∂1F(x0, y0) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition on F as a function of x. © 1978 Plenum Publishing Corporation.