If $f(x)$ is a continously differentiable function such that $f'(x)$ is unbounded, then $f(x)$ is not...












2












$begingroup$


Suppose $f(x)$ is a continuously differentiable function on $mathbb{R}$ such that its derivative is unbounded.



Claim: $f(x)$ is not uniformly continuous.



Assume f(x) is uniformly continuous, then given $epsilon$ we have a suitable $delta$. Since $f'(x)$ is not bounded, their exists an element $c$ such that $f'(c) > k frac{epsilon}{delta}$, $k gg 0$. Since $f'(x)$ is continuous, if we choose a $delta$ neighbourhood of $c$, we would then by mean value theorem have
$$f(b) - f(a) = f'(c')(b-a), c' in Big(c- frac{delta}{2},; c + frac{delta}{2}Big).$$ or



$$ f'(c') = frac{epsilon}{delta}$$ - a contradiction since $f'(c')$ has value in around some small neighborhood of $k frac{epsilon}{delta}$.



So finally I have this conclusion:




If $f(x)$ is continuously differential function, then $f(x)$ is uniformly continuous iff $f'(x)$ is bounded?




Is my conclusion right?



New details:



I realize that that my argument $f'(c) to f'(c')$ is slippery and might not yield the desired conclusion every time, but suppose if I have a specific $f(x)$ and I can somehow show that $m leq mid f'(x) mid leq M$ for some interval $I$ where $f'(c) = kfrac{epsilon}{delta}$ and $m, M$ are suitable close enough (say $epsilon^{1000000})$, then does my argument holds?










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Suppose $f(x)$ is a continuously differentiable function on $mathbb{R}$ such that its derivative is unbounded.



    Claim: $f(x)$ is not uniformly continuous.



    Assume f(x) is uniformly continuous, then given $epsilon$ we have a suitable $delta$. Since $f'(x)$ is not bounded, their exists an element $c$ such that $f'(c) > k frac{epsilon}{delta}$, $k gg 0$. Since $f'(x)$ is continuous, if we choose a $delta$ neighbourhood of $c$, we would then by mean value theorem have
    $$f(b) - f(a) = f'(c')(b-a), c' in Big(c- frac{delta}{2},; c + frac{delta}{2}Big).$$ or



    $$ f'(c') = frac{epsilon}{delta}$$ - a contradiction since $f'(c')$ has value in around some small neighborhood of $k frac{epsilon}{delta}$.



    So finally I have this conclusion:




    If $f(x)$ is continuously differential function, then $f(x)$ is uniformly continuous iff $f'(x)$ is bounded?




    Is my conclusion right?



    New details:



    I realize that that my argument $f'(c) to f'(c')$ is slippery and might not yield the desired conclusion every time, but suppose if I have a specific $f(x)$ and I can somehow show that $m leq mid f'(x) mid leq M$ for some interval $I$ where $f'(c) = kfrac{epsilon}{delta}$ and $m, M$ are suitable close enough (say $epsilon^{1000000})$, then does my argument holds?










    share|cite|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      Suppose $f(x)$ is a continuously differentiable function on $mathbb{R}$ such that its derivative is unbounded.



      Claim: $f(x)$ is not uniformly continuous.



      Assume f(x) is uniformly continuous, then given $epsilon$ we have a suitable $delta$. Since $f'(x)$ is not bounded, their exists an element $c$ such that $f'(c) > k frac{epsilon}{delta}$, $k gg 0$. Since $f'(x)$ is continuous, if we choose a $delta$ neighbourhood of $c$, we would then by mean value theorem have
      $$f(b) - f(a) = f'(c')(b-a), c' in Big(c- frac{delta}{2},; c + frac{delta}{2}Big).$$ or



      $$ f'(c') = frac{epsilon}{delta}$$ - a contradiction since $f'(c')$ has value in around some small neighborhood of $k frac{epsilon}{delta}$.



      So finally I have this conclusion:




      If $f(x)$ is continuously differential function, then $f(x)$ is uniformly continuous iff $f'(x)$ is bounded?




      Is my conclusion right?



      New details:



      I realize that that my argument $f'(c) to f'(c')$ is slippery and might not yield the desired conclusion every time, but suppose if I have a specific $f(x)$ and I can somehow show that $m leq mid f'(x) mid leq M$ for some interval $I$ where $f'(c) = kfrac{epsilon}{delta}$ and $m, M$ are suitable close enough (say $epsilon^{1000000})$, then does my argument holds?










      share|cite|improve this question











      $endgroup$




      Suppose $f(x)$ is a continuously differentiable function on $mathbb{R}$ such that its derivative is unbounded.



      Claim: $f(x)$ is not uniformly continuous.



      Assume f(x) is uniformly continuous, then given $epsilon$ we have a suitable $delta$. Since $f'(x)$ is not bounded, their exists an element $c$ such that $f'(c) > k frac{epsilon}{delta}$, $k gg 0$. Since $f'(x)$ is continuous, if we choose a $delta$ neighbourhood of $c$, we would then by mean value theorem have
      $$f(b) - f(a) = f'(c')(b-a), c' in Big(c- frac{delta}{2},; c + frac{delta}{2}Big).$$ or



      $$ f'(c') = frac{epsilon}{delta}$$ - a contradiction since $f'(c')$ has value in around some small neighborhood of $k frac{epsilon}{delta}$.



      So finally I have this conclusion:




      If $f(x)$ is continuously differential function, then $f(x)$ is uniformly continuous iff $f'(x)$ is bounded?




      Is my conclusion right?



      New details:



      I realize that that my argument $f'(c) to f'(c')$ is slippery and might not yield the desired conclusion every time, but suppose if I have a specific $f(x)$ and I can somehow show that $m leq mid f'(x) mid leq M$ for some interval $I$ where $f'(c) = kfrac{epsilon}{delta}$ and $m, M$ are suitable close enough (say $epsilon^{1000000})$, then does my argument holds?







      uniform-continuity






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      edited Dec 6 '18 at 12:49







      henceproved

















      asked Nov 30 '18 at 9:17









      henceprovedhenceproved

      1358




      1358






















          3 Answers
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          $begingroup$

          Take any unbounded continuous integrable function $g$ and define $f(x)=int_0^{x} g(y), dy$. Then $f$ is uniformly continuous but its derivative is not bounded. To construct such a function $g$ draw triangles with bases $(n-frac 1 {n^{3}}, n+frac 1 {n^{3}})$ and height $n$. Think of a function $g$ whose graph is made up of these triangles. $f$ is uniformly continuous because it is continuous and it has finite limits at $pm infty$.






          share|cite|improve this answer











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            $begingroup$

            The error in OP's argument is that they fix $epsilon, delta$ beforehand for the uniformity claim, but later use the fact that the derivative $f'$ is above a certain value in an interval of length $delta$, which is not guaranteed, as Kavi explained.



            Intuitively, if the interval where $f'$ is high is very small, and becomes smaller with higher $f'$, for a given $epsilon$ the 'spikes' in $f'$ (as described by Kavi) are not able to increase $f$ by much (more than $epsilon$) except for a finite number at the beginning.






            share|cite|improve this answer









            $endgroup$





















              1












              $begingroup$

              No, you are not right. Consider for example the $C^{infty}$ function
              $$f(x)=frac{cos(e^x)}{1+x^2}.$$
              $f$ is uniformly continuous in $mathbb{R}$ because it is continuous and its limit at $pm infty$ is zero.



              On the other hand, its derivative is unbounded in $mathbb{R}$:
              $$f'(x)=-frac{e^xsin(e^x)}{1+x^2}-frac{2xcos(e^x)}{(1+x^2)^2}$$
              and $lim_{nto +infty} f'(x_n)=+infty$ where $x_n=ln(3pi/2+2pi n)$.






              share|cite|improve this answer











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                3 Answers
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                $begingroup$

                Take any unbounded continuous integrable function $g$ and define $f(x)=int_0^{x} g(y), dy$. Then $f$ is uniformly continuous but its derivative is not bounded. To construct such a function $g$ draw triangles with bases $(n-frac 1 {n^{3}}, n+frac 1 {n^{3}})$ and height $n$. Think of a function $g$ whose graph is made up of these triangles. $f$ is uniformly continuous because it is continuous and it has finite limits at $pm infty$.






                share|cite|improve this answer











                $endgroup$


















                  4












                  $begingroup$

                  Take any unbounded continuous integrable function $g$ and define $f(x)=int_0^{x} g(y), dy$. Then $f$ is uniformly continuous but its derivative is not bounded. To construct such a function $g$ draw triangles with bases $(n-frac 1 {n^{3}}, n+frac 1 {n^{3}})$ and height $n$. Think of a function $g$ whose graph is made up of these triangles. $f$ is uniformly continuous because it is continuous and it has finite limits at $pm infty$.






                  share|cite|improve this answer











                  $endgroup$
















                    4












                    4








                    4





                    $begingroup$

                    Take any unbounded continuous integrable function $g$ and define $f(x)=int_0^{x} g(y), dy$. Then $f$ is uniformly continuous but its derivative is not bounded. To construct such a function $g$ draw triangles with bases $(n-frac 1 {n^{3}}, n+frac 1 {n^{3}})$ and height $n$. Think of a function $g$ whose graph is made up of these triangles. $f$ is uniformly continuous because it is continuous and it has finite limits at $pm infty$.






                    share|cite|improve this answer











                    $endgroup$



                    Take any unbounded continuous integrable function $g$ and define $f(x)=int_0^{x} g(y), dy$. Then $f$ is uniformly continuous but its derivative is not bounded. To construct such a function $g$ draw triangles with bases $(n-frac 1 {n^{3}}, n+frac 1 {n^{3}})$ and height $n$. Think of a function $g$ whose graph is made up of these triangles. $f$ is uniformly continuous because it is continuous and it has finite limits at $pm infty$.







                    share|cite|improve this answer














                    share|cite|improve this answer



                    share|cite|improve this answer








                    edited Nov 30 '18 at 9:29

























                    answered Nov 30 '18 at 9:23









                    Kavi Rama MurthyKavi Rama Murthy

                    52.6k32055




                    52.6k32055























                        1












                        $begingroup$

                        The error in OP's argument is that they fix $epsilon, delta$ beforehand for the uniformity claim, but later use the fact that the derivative $f'$ is above a certain value in an interval of length $delta$, which is not guaranteed, as Kavi explained.



                        Intuitively, if the interval where $f'$ is high is very small, and becomes smaller with higher $f'$, for a given $epsilon$ the 'spikes' in $f'$ (as described by Kavi) are not able to increase $f$ by much (more than $epsilon$) except for a finite number at the beginning.






                        share|cite|improve this answer









                        $endgroup$


















                          1












                          $begingroup$

                          The error in OP's argument is that they fix $epsilon, delta$ beforehand for the uniformity claim, but later use the fact that the derivative $f'$ is above a certain value in an interval of length $delta$, which is not guaranteed, as Kavi explained.



                          Intuitively, if the interval where $f'$ is high is very small, and becomes smaller with higher $f'$, for a given $epsilon$ the 'spikes' in $f'$ (as described by Kavi) are not able to increase $f$ by much (more than $epsilon$) except for a finite number at the beginning.






                          share|cite|improve this answer









                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            The error in OP's argument is that they fix $epsilon, delta$ beforehand for the uniformity claim, but later use the fact that the derivative $f'$ is above a certain value in an interval of length $delta$, which is not guaranteed, as Kavi explained.



                            Intuitively, if the interval where $f'$ is high is very small, and becomes smaller with higher $f'$, for a given $epsilon$ the 'spikes' in $f'$ (as described by Kavi) are not able to increase $f$ by much (more than $epsilon$) except for a finite number at the beginning.






                            share|cite|improve this answer









                            $endgroup$



                            The error in OP's argument is that they fix $epsilon, delta$ beforehand for the uniformity claim, but later use the fact that the derivative $f'$ is above a certain value in an interval of length $delta$, which is not guaranteed, as Kavi explained.



                            Intuitively, if the interval where $f'$ is high is very small, and becomes smaller with higher $f'$, for a given $epsilon$ the 'spikes' in $f'$ (as described by Kavi) are not able to increase $f$ by much (more than $epsilon$) except for a finite number at the beginning.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 30 '18 at 9:38









                            IngixIngix

                            3,329146




                            3,329146























                                1












                                $begingroup$

                                No, you are not right. Consider for example the $C^{infty}$ function
                                $$f(x)=frac{cos(e^x)}{1+x^2}.$$
                                $f$ is uniformly continuous in $mathbb{R}$ because it is continuous and its limit at $pm infty$ is zero.



                                On the other hand, its derivative is unbounded in $mathbb{R}$:
                                $$f'(x)=-frac{e^xsin(e^x)}{1+x^2}-frac{2xcos(e^x)}{(1+x^2)^2}$$
                                and $lim_{nto +infty} f'(x_n)=+infty$ where $x_n=ln(3pi/2+2pi n)$.






                                share|cite|improve this answer











                                $endgroup$


















                                  1












                                  $begingroup$

                                  No, you are not right. Consider for example the $C^{infty}$ function
                                  $$f(x)=frac{cos(e^x)}{1+x^2}.$$
                                  $f$ is uniformly continuous in $mathbb{R}$ because it is continuous and its limit at $pm infty$ is zero.



                                  On the other hand, its derivative is unbounded in $mathbb{R}$:
                                  $$f'(x)=-frac{e^xsin(e^x)}{1+x^2}-frac{2xcos(e^x)}{(1+x^2)^2}$$
                                  and $lim_{nto +infty} f'(x_n)=+infty$ where $x_n=ln(3pi/2+2pi n)$.






                                  share|cite|improve this answer











                                  $endgroup$
















                                    1












                                    1








                                    1





                                    $begingroup$

                                    No, you are not right. Consider for example the $C^{infty}$ function
                                    $$f(x)=frac{cos(e^x)}{1+x^2}.$$
                                    $f$ is uniformly continuous in $mathbb{R}$ because it is continuous and its limit at $pm infty$ is zero.



                                    On the other hand, its derivative is unbounded in $mathbb{R}$:
                                    $$f'(x)=-frac{e^xsin(e^x)}{1+x^2}-frac{2xcos(e^x)}{(1+x^2)^2}$$
                                    and $lim_{nto +infty} f'(x_n)=+infty$ where $x_n=ln(3pi/2+2pi n)$.






                                    share|cite|improve this answer











                                    $endgroup$



                                    No, you are not right. Consider for example the $C^{infty}$ function
                                    $$f(x)=frac{cos(e^x)}{1+x^2}.$$
                                    $f$ is uniformly continuous in $mathbb{R}$ because it is continuous and its limit at $pm infty$ is zero.



                                    On the other hand, its derivative is unbounded in $mathbb{R}$:
                                    $$f'(x)=-frac{e^xsin(e^x)}{1+x^2}-frac{2xcos(e^x)}{(1+x^2)^2}$$
                                    and $lim_{nto +infty} f'(x_n)=+infty$ where $x_n=ln(3pi/2+2pi n)$.







                                    share|cite|improve this answer














                                    share|cite|improve this answer



                                    share|cite|improve this answer








                                    edited Nov 30 '18 at 16:10

























                                    answered Nov 30 '18 at 9:28









                                    Robert ZRobert Z

                                    94.2k1061132




                                    94.2k1061132






























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