If derivative of $f$ is continuous, then $f$ is continuous.












5












$begingroup$


I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of




If $f$ is differentiable, then $f$ is continuous




but it's not the same argument I think. Also, I want to know what's your opinion about the statement




If derivative of $f$ is not continuous, then $f$ is not continuous







calculus derivatives continuity







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Continuous at R or at domain of a function ?
    $endgroup$
    – Юрій Ярош
    Apr 17 '18 at 11:35










  • $begingroup$
    1. Continuous at R 2. Continuous at a spesific point x0.
    $endgroup$
    – Alex Spanos
    Apr 17 '18 at 11:39






  • 13




    $begingroup$
    "what's your opinion ..." Math isn't about opinions.
    $endgroup$
    – John Coleman
    Apr 17 '18 at 13:30








  • 4




    $begingroup$
    "If a car is red, then it is a vehicle."
    $endgroup$
    – Eric Duminil
    Apr 18 '18 at 8:04












  • $begingroup$
    @JohnColeman It is not about, however there is room for them. How about open problems?
    $endgroup$
    – TStancek
    Apr 18 '18 at 9:32


















5












$begingroup$


I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of




If $f$ is differentiable, then $f$ is continuous




but it's not the same argument I think. Also, I want to know what's your opinion about the statement




If derivative of $f$ is not continuous, then $f$ is not continuous







calculus derivatives continuity







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Continuous at R or at domain of a function ?
    $endgroup$
    – Юрій Ярош
    Apr 17 '18 at 11:35










  • $begingroup$
    1. Continuous at R 2. Continuous at a spesific point x0.
    $endgroup$
    – Alex Spanos
    Apr 17 '18 at 11:39






  • 13




    $begingroup$
    "what's your opinion ..." Math isn't about opinions.
    $endgroup$
    – John Coleman
    Apr 17 '18 at 13:30








  • 4




    $begingroup$
    "If a car is red, then it is a vehicle."
    $endgroup$
    – Eric Duminil
    Apr 18 '18 at 8:04












  • $begingroup$
    @JohnColeman It is not about, however there is room for them. How about open problems?
    $endgroup$
    – TStancek
    Apr 18 '18 at 9:32
















5












5








5


1



$begingroup$


I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of




If $f$ is differentiable, then $f$ is continuous




but it's not the same argument I think. Also, I want to know what's your opinion about the statement




If derivative of $f$ is not continuous, then $f$ is not continuous







calculus derivatives continuity







share|cite|improve this question











$endgroup$




I have searched a lot, but i haven't found any proof about that statement. I have checked the proof of




If $f$ is differentiable, then $f$ is continuous




but it's not the same argument I think. Also, I want to know what's your opinion about the statement




If derivative of $f$ is not continuous, then $f$ is not continuous







calculus derivatives continuity




calculus derivatives continuity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 29 '18 at 20:34









José Carlos Santos

169k23132237




169k23132237










asked Apr 17 '18 at 11:31









Alex SpanosAlex Spanos

3817




3817








  • 1




    $begingroup$
    Continuous at R or at domain of a function ?
    $endgroup$
    – Юрій Ярош
    Apr 17 '18 at 11:35










  • $begingroup$
    1. Continuous at R 2. Continuous at a spesific point x0.
    $endgroup$
    – Alex Spanos
    Apr 17 '18 at 11:39






  • 13




    $begingroup$
    "what's your opinion ..." Math isn't about opinions.
    $endgroup$
    – John Coleman
    Apr 17 '18 at 13:30








  • 4




    $begingroup$
    "If a car is red, then it is a vehicle."
    $endgroup$
    – Eric Duminil
    Apr 18 '18 at 8:04












  • $begingroup$
    @JohnColeman It is not about, however there is room for them. How about open problems?
    $endgroup$
    – TStancek
    Apr 18 '18 at 9:32
















  • 1




    $begingroup$
    Continuous at R or at domain of a function ?
    $endgroup$
    – Юрій Ярош
    Apr 17 '18 at 11:35










  • $begingroup$
    1. Continuous at R 2. Continuous at a spesific point x0.
    $endgroup$
    – Alex Spanos
    Apr 17 '18 at 11:39






  • 13




    $begingroup$
    "what's your opinion ..." Math isn't about opinions.
    $endgroup$
    – John Coleman
    Apr 17 '18 at 13:30








  • 4




    $begingroup$
    "If a car is red, then it is a vehicle."
    $endgroup$
    – Eric Duminil
    Apr 18 '18 at 8:04












  • $begingroup$
    @JohnColeman It is not about, however there is room for them. How about open problems?
    $endgroup$
    – TStancek
    Apr 18 '18 at 9:32










1




1




$begingroup$
Continuous at R or at domain of a function ?
$endgroup$
– Юрій Ярош
Apr 17 '18 at 11:35




$begingroup$
Continuous at R or at domain of a function ?
$endgroup$
– Юрій Ярош
Apr 17 '18 at 11:35












$begingroup$
1. Continuous at R 2. Continuous at a spesific point x0.
$endgroup$
– Alex Spanos
Apr 17 '18 at 11:39




$begingroup$
1. Continuous at R 2. Continuous at a spesific point x0.
$endgroup$
– Alex Spanos
Apr 17 '18 at 11:39




13




13




$begingroup$
"what's your opinion ..." Math isn't about opinions.
$endgroup$
– John Coleman
Apr 17 '18 at 13:30






$begingroup$
"what's your opinion ..." Math isn't about opinions.
$endgroup$
– John Coleman
Apr 17 '18 at 13:30






4




4




$begingroup$
"If a car is red, then it is a vehicle."
$endgroup$
– Eric Duminil
Apr 18 '18 at 8:04






$begingroup$
"If a car is red, then it is a vehicle."
$endgroup$
– Eric Duminil
Apr 18 '18 at 8:04














$begingroup$
@JohnColeman It is not about, however there is room for them. How about open problems?
$endgroup$
– TStancek
Apr 18 '18 at 9:32






$begingroup$
@JohnColeman It is not about, however there is room for them. How about open problems?
$endgroup$
– TStancek
Apr 18 '18 at 9:32












4 Answers
4






active

oldest

votes


















41












$begingroup$

If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.



In particular, even if $f'$ is discontinuous, $f$ is continuous.






share|cite|improve this answer









$endgroup$









  • 4




    $begingroup$
    Note: the abovewritten is true if we're not taking generalized functions into account
    $endgroup$
    – Dmitry Ginzburg
    Apr 17 '18 at 14:09



















14












$begingroup$

Your problem seems to be the logical relationships between the statements





  1. If f is differentiable, then it is continuous

  2. If the derivative of $f$ is continuous, then $f$ is continuous


  3. If the derivative of $f$ is not continuous, then $f$ is not continous.


The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{prime}$ is continuous".



The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function
$$f(x)=x^2sinleft(frac{1}{x}right)$$
is differentiable everywhere, with derivative
$$f^{prime}(x)=left{begin{array}{ll}
2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)& xneq 0
\
0 & x=0
end{array}right.$$
But $lim_{xto 0}f^{prime}(x)$ does not exist, hence $f^{prime}$ is not continuous.






share|cite|improve this answer











$endgroup$





















    2












    $begingroup$

    $f'$ need not be continuous.



    Suppose that $f'(x)$ exists in the interval $(a,b)$. If $xi in (a,b)$, then $f'(xi)$ exists. Hence $f$ is continuous at $xi$. Since this is true for all $xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Sorry. It should have said $f'$ need not be continuous.
      $endgroup$
      – steven gregory
      Apr 17 '18 at 17:24



















    0












    $begingroup$

    Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable.



    This does not disprove the opposite statement, of course.






    share|cite|improve this answer









    $endgroup$













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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      41












      $begingroup$

      If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.



      In particular, even if $f'$ is discontinuous, $f$ is continuous.






      share|cite|improve this answer









      $endgroup$









      • 4




        $begingroup$
        Note: the abovewritten is true if we're not taking generalized functions into account
        $endgroup$
        – Dmitry Ginzburg
        Apr 17 '18 at 14:09
















      41












      $begingroup$

      If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.



      In particular, even if $f'$ is discontinuous, $f$ is continuous.






      share|cite|improve this answer









      $endgroup$









      • 4




        $begingroup$
        Note: the abovewritten is true if we're not taking generalized functions into account
        $endgroup$
        – Dmitry Ginzburg
        Apr 17 '18 at 14:09














      41












      41








      41





      $begingroup$

      If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.



      In particular, even if $f'$ is discontinuous, $f$ is continuous.






      share|cite|improve this answer









      $endgroup$



      If $f$ is differentiable, then $f$ is continuous. The continuity of $f'$ is irrelevant here.



      In particular, even if $f'$ is discontinuous, $f$ is continuous.







      share|cite|improve this answer












      share|cite|improve this answer



      share|cite|improve this answer










      answered Apr 17 '18 at 11:32









      José Carlos SantosJosé Carlos Santos

      169k23132237




      169k23132237








      • 4




        $begingroup$
        Note: the abovewritten is true if we're not taking generalized functions into account
        $endgroup$
        – Dmitry Ginzburg
        Apr 17 '18 at 14:09














      • 4




        $begingroup$
        Note: the abovewritten is true if we're not taking generalized functions into account
        $endgroup$
        – Dmitry Ginzburg
        Apr 17 '18 at 14:09








      4




      4




      $begingroup$
      Note: the abovewritten is true if we're not taking generalized functions into account
      $endgroup$
      – Dmitry Ginzburg
      Apr 17 '18 at 14:09




      $begingroup$
      Note: the abovewritten is true if we're not taking generalized functions into account
      $endgroup$
      – Dmitry Ginzburg
      Apr 17 '18 at 14:09











      14












      $begingroup$

      Your problem seems to be the logical relationships between the statements





      1. If f is differentiable, then it is continuous

      2. If the derivative of $f$ is continuous, then $f$ is continuous


      3. If the derivative of $f$ is not continuous, then $f$ is not continous.


      The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{prime}$ is continuous".



      The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function
      $$f(x)=x^2sinleft(frac{1}{x}right)$$
      is differentiable everywhere, with derivative
      $$f^{prime}(x)=left{begin{array}{ll}
      2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)& xneq 0
      \
      0 & x=0
      end{array}right.$$
      But $lim_{xto 0}f^{prime}(x)$ does not exist, hence $f^{prime}$ is not continuous.






      share|cite|improve this answer











      $endgroup$


















        14












        $begingroup$

        Your problem seems to be the logical relationships between the statements





        1. If f is differentiable, then it is continuous

        2. If the derivative of $f$ is continuous, then $f$ is continuous


        3. If the derivative of $f$ is not continuous, then $f$ is not continous.


        The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{prime}$ is continuous".



        The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function
        $$f(x)=x^2sinleft(frac{1}{x}right)$$
        is differentiable everywhere, with derivative
        $$f^{prime}(x)=left{begin{array}{ll}
        2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)& xneq 0
        \
        0 & x=0
        end{array}right.$$
        But $lim_{xto 0}f^{prime}(x)$ does not exist, hence $f^{prime}$ is not continuous.






        share|cite|improve this answer











        $endgroup$
















          14












          14








          14





          $begingroup$

          Your problem seems to be the logical relationships between the statements





          1. If f is differentiable, then it is continuous

          2. If the derivative of $f$ is continuous, then $f$ is continuous


          3. If the derivative of $f$ is not continuous, then $f$ is not continous.


          The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{prime}$ is continuous".



          The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function
          $$f(x)=x^2sinleft(frac{1}{x}right)$$
          is differentiable everywhere, with derivative
          $$f^{prime}(x)=left{begin{array}{ll}
          2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)& xneq 0
          \
          0 & x=0
          end{array}right.$$
          But $lim_{xto 0}f^{prime}(x)$ does not exist, hence $f^{prime}$ is not continuous.






          share|cite|improve this answer











          $endgroup$



          Your problem seems to be the logical relationships between the statements





          1. If f is differentiable, then it is continuous

          2. If the derivative of $f$ is continuous, then $f$ is continuous


          3. If the derivative of $f$ is not continuous, then $f$ is not continous.


          The first statement trivially implies the second, since saying "the derivative of $f$ is continuous" is the same as saying "$f$ is differentiable and $f^{prime}$ is continuous".



          The contrapositive of the third statement is "If $f$ is continuous, then the derivative of $f$ is continuous." This is false. For example, the function
          $$f(x)=x^2sinleft(frac{1}{x}right)$$
          is differentiable everywhere, with derivative
          $$f^{prime}(x)=left{begin{array}{ll}
          2xsinleft(frac{1}{x}right)-cosleft(frac{1}{x}right)& xneq 0
          \
          0 & x=0
          end{array}right.$$
          But $lim_{xto 0}f^{prime}(x)$ does not exist, hence $f^{prime}$ is not continuous.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 30 '18 at 15:01









          José Carlos Santos

          169k23132237




          169k23132237










          answered Apr 17 '18 at 12:15









          user43687user43687

          2,643816




          2,643816























              2












              $begingroup$

              $f'$ need not be continuous.



              Suppose that $f'(x)$ exists in the interval $(a,b)$. If $xi in (a,b)$, then $f'(xi)$ exists. Hence $f$ is continuous at $xi$. Since this is true for all $xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                Sorry. It should have said $f'$ need not be continuous.
                $endgroup$
                – steven gregory
                Apr 17 '18 at 17:24
















              2












              $begingroup$

              $f'$ need not be continuous.



              Suppose that $f'(x)$ exists in the interval $(a,b)$. If $xi in (a,b)$, then $f'(xi)$ exists. Hence $f$ is continuous at $xi$. Since this is true for all $xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.






              share|cite|improve this answer











              $endgroup$









              • 1




                $begingroup$
                Sorry. It should have said $f'$ need not be continuous.
                $endgroup$
                – steven gregory
                Apr 17 '18 at 17:24














              2












              2








              2





              $begingroup$

              $f'$ need not be continuous.



              Suppose that $f'(x)$ exists in the interval $(a,b)$. If $xi in (a,b)$, then $f'(xi)$ exists. Hence $f$ is continuous at $xi$. Since this is true for all $xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.






              share|cite|improve this answer











              $endgroup$



              $f'$ need not be continuous.



              Suppose that $f'(x)$ exists in the interval $(a,b)$. If $xi in (a,b)$, then $f'(xi)$ exists. Hence $f$ is continuous at $xi$. Since this is true for all $xi$ in $(a,b)$, then $f$ is continuous on $(a,b)$.







              share|cite|improve this answer














              share|cite|improve this answer



              share|cite|improve this answer








              edited Apr 17 '18 at 17:23

























              answered Apr 17 '18 at 15:40









              steven gregorysteven gregory

              18.3k32358




              18.3k32358








              • 1




                $begingroup$
                Sorry. It should have said $f'$ need not be continuous.
                $endgroup$
                – steven gregory
                Apr 17 '18 at 17:24














              • 1




                $begingroup$
                Sorry. It should have said $f'$ need not be continuous.
                $endgroup$
                – steven gregory
                Apr 17 '18 at 17:24








              1




              1




              $begingroup$
              Sorry. It should have said $f'$ need not be continuous.
              $endgroup$
              – steven gregory
              Apr 17 '18 at 17:24




              $begingroup$
              Sorry. It should have said $f'$ need not be continuous.
              $endgroup$
              – steven gregory
              Apr 17 '18 at 17:24











              0












              $begingroup$

              Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable.



              This does not disprove the opposite statement, of course.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable.



                This does not disprove the opposite statement, of course.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable.



                  This does not disprove the opposite statement, of course.






                  share|cite|improve this answer









                  $endgroup$



                  Answering only part of the question: "If derivative of f is not continuous, then f is not continuous": As perhaps the simplest counter-example, the absolute value function is continuous but not continuously differentiable.



                  This does not disprove the opposite statement, of course.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Apr 18 '18 at 11:24









                  arparp

                  1012




                  1012






























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