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Some difficulties in understanding the proof of the inverse function theorem.



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)A difficulty in understanding the proof of completeness of $l_2$.A difficulty in understanding theorem 4.2 in Israel Gohberg.A difficulty in understanding the proof of theorem 1.14 in Hungerford.Second difficulty in understanding the proof of theorem 1.14 in Hungerford.A difficulty in obtaining a formula in the proof of a theorem.A difficulty in understanding a proof for L'Hospital's rule (in Petrovic)A difficulty in understanding theorem 10.6.7 in Petrovic.(n-dimensional intermediate value theorem)A difficulty in understanding a step in the proof of Thm. 11.5.6 in Petrovic.A difficulty in understanding multivariable Fermat theorem proof.A difficulty in understanding the proof of the open mapping theorem.










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$begingroup$


The theorem and a part of its proof is given below:




enter image description hereenter image description hereenter image description hereenter image description here




but it is not clear for me how he proved (d), could anyone explain this for me please?



Also in this last part of the proof:




enter image description here




why can we take $r_h(y) = r_f(x)$ ?










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    The theorem and a part of its proof is given below:




    enter image description hereenter image description hereenter image description hereenter image description here




    but it is not clear for me how he proved (d), could anyone explain this for me please?



    Also in this last part of the proof:




    enter image description here




    why can we take $r_h(y) = r_f(x)$ ?










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      The theorem and a part of its proof is given below:




      enter image description hereenter image description hereenter image description hereenter image description here




      but it is not clear for me how he proved (d), could anyone explain this for me please?



      Also in this last part of the proof:




      enter image description here




      why can we take $r_h(y) = r_f(x)$ ?










      share|cite|improve this question











      $endgroup$




      The theorem and a part of its proof is given below:




      enter image description hereenter image description hereenter image description hereenter image description here




      but it is not clear for me how he proved (d), could anyone explain this for me please?



      Also in this last part of the proof:




      enter image description here




      why can we take $r_h(y) = r_f(x)$ ?







      real-analysis calculus multivariable-calculus proof-explanation inverse-function-theorem






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Apr 2 at 15:40







      Smart

















      asked Apr 2 at 15:22









      SmartSmart

      948




      948




















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