An implicit function theorem: Comment

  title={An implicit function theorem: Comment},
  author={Sadatoshi Kumagai},
  journal={Journal of Optimization Theory and Applications},
  • S. Kumagai
  • Published 1 June 1980
  • Mathematics
  • Journal of Optimization Theory and Applications
In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:Rn ×Rm →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… Expand
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