An implicit function theorem: Comment

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:Rⁿ ×R^m Rⁿ, withF(x⁰,y⁰)=0, that requires neither differentiability ofF nor nonsingularity of _xF(x⁰,y⁰). In the proof, the local one-to-one condition forF(·,y):A Rⁿ Rⁿ 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.