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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                      Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
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                      Dieringhausen

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                      Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
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