An implicit function theorem: Comment

Abstract

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:R ⁿ ×R ^m →R ⁿ, withF(x ⁰,y ⁰)=0, that requires neither differentiability ofF nor nonsingularity of ∂ _x F(x ⁰,y ⁰). In the proof, the local one-to-one condition forF(·,y):A ⊂R ⁿ →R ⁿ for ally ∈B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally ∈B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that the condition is truly if and only if.