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We formulate some global invertibility and implicit function theorems. Weextend the result of Idczak, Skowron and Walczak on the invertibility of theoperators to the case of the operators with critical points. The proof relieson the Mountain Pass Theorem combined with the Palais-Smale conditionguaranteeing the claim by the invertibility of the first or the thirdderivative. I. e. how to solve $x^3=y$?

Global invertibility and implicit function theorems by mountain pass theorem