Let $f:Mrightarrow mathbb R$ be continuously differentiable with $nabla f(x)=0$ for all $xin M$, then $f$ is constant

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Let $M:=(x,y)in mathbb R^2:x^2+y^2<1setminus(x,0)in mathbb R^2, xin mathbb R$ and $f:Mrightarrow mathbb R$ be continuously differentiable with $nabla f(x)=0$ for all $xin M$, then $f$ is constant.



I guess that this statement is true, but I don't know how to prove it. I guess the mean-value theorem can help here?




real-analysis




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    Let $M:=(x,y)in mathbb R^2:x^2+y^2<1setminus(x,0)in mathbb R^2, xin mathbb R$ and $f:Mrightarrow mathbb R$ be continuously differentiable with $nabla f(x)=0$ for all $xin M$, then $f$ is constant.



    I guess that this statement is true, but I don't know how to prove it. I guess the mean-value theorem can help here?




    real-analysis




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      up vote
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      down vote

      favorite









      up vote
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      down vote

      favorite











      Let $M:=(x,y)in mathbb R^2:x^2+y^2<1setminus(x,0)in mathbb R^2, xin mathbb R$ and $f:Mrightarrow mathbb R$ be continuously differentiable with $nabla f(x)=0$ for all $xin M$, then $f$ is constant.



      I guess that this statement is true, but I don't know how to prove it. I guess the mean-value theorem can help here?




      real-analysis




      share|cite|improve this question











      Let $M:=(x,y)in mathbb R^2:x^2+y^2<1setminus(x,0)in mathbb R^2, xin mathbb R$ and $f:Mrightarrow mathbb R$ be continuously differentiable with $nabla f(x)=0$ for all $xin M$, then $f$ is constant.



      I guess that this statement is true, but I don't know how to prove it. I guess the mean-value theorem can help here?





      real-analysis





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      asked 20 hours ago









      Marc

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          Draw a picture and you see that $M$ is two "half circles" which is not connected. Let $f$ have different constant value on every half circle to get a counterexemple.






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            No, it is not true. Take$$f(x,y)=begincases1&text if y>0\0&text otherwise.endcases$$






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              2 Answers
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              2 Answers
              2






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              up vote
              3
              down vote



              accepted










              Draw a picture and you see that $M$ is two "half circles" which is not connected. Let $f$ have different constant value on every half circle to get a counterexemple.






              share|cite|improve this answer

























                up vote
                3
                down vote



                accepted










                Draw a picture and you see that $M$ is two "half circles" which is not connected. Let $f$ have different constant value on every half circle to get a counterexemple.






                share|cite|improve this answer























                  up vote
                  3
                  down vote



                  accepted







                  up vote
                  3
                  down vote



                  accepted






                  Draw a picture and you see that $M$ is two "half circles" which is not connected. Let $f$ have different constant value on every half circle to get a counterexemple.






                  share|cite|improve this answer













                  Draw a picture and you see that $M$ is two "half circles" which is not connected. Let $f$ have different constant value on every half circle to get a counterexemple.







                  share|cite|improve this answer













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                  answered 20 hours ago









                  user296113

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                      No, it is not true. Take$$f(x,y)=begincases1&text if y>0\0&text otherwise.endcases$$






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                        up vote
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                        No, it is not true. Take$$f(x,y)=begincases1&text if y>0\0&text otherwise.endcases$$






                        share|cite|improve this answer























                          up vote
                          1
                          down vote










                          up vote
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                          down vote









                          No, it is not true. Take$$f(x,y)=begincases1&text if y>0\0&text otherwise.endcases$$






                          share|cite|improve this answer













                          No, it is not true. Take$$f(x,y)=begincases1&text if y>0\0&text otherwise.endcases$$







                          share|cite|improve this answer













                          share|cite|improve this answer



                          share|cite|improve this answer











                          answered 20 hours ago









                          José Carlos Santos

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