compactly support function with constant value












1












$begingroup$


I am interested in finding a function $varphicolonmathbb{R}tomathbb{R}$ with the following property:





  • $varphi$ has compact support, which contains $left[0,1right]$


  • $varphi$ is continuously differentiable on $mathbb{R}$ at every order.


  • $varphi$ is constant on (at least one) an intevarl of the form $left[0,1/nright]$, where $ninmathbb{N}$.


As pointed out in an exercise in the book "Functional Analysis, Sobolev Spaces, and Differential Equations" by Haim Brezis, we can construct one via mollifiers. However, it does not seem to give a closed form or explicit formula that we can easily play with.



So here is my question: Is there an example where we have a closed form that is easy to play with?



Any help/hint is highly appreciated.






real-analysis integration functional-analysis derivatives distribution-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    This might be helpful.
    $endgroup$
    – MisterRiemann
    Nov 30 '18 at 18:33






  • 2




    $begingroup$
    You mean *mollifiers, not modifiers, presumably
    $endgroup$
    – qbert
    Nov 30 '18 at 18:37






  • 1




    $begingroup$
    I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
    $endgroup$
    – Clayton
    Nov 30 '18 at 18:38










  • $begingroup$
    @MisterRiemann Yes, very helpful indeed.
    $endgroup$
    – weirdo
    Nov 30 '18 at 18:38






  • 2




    $begingroup$
    Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
    $endgroup$
    – reuns
    Nov 30 '18 at 18:47


















1












$begingroup$


I am interested in finding a function $varphicolonmathbb{R}tomathbb{R}$ with the following property:





  • $varphi$ has compact support, which contains $left[0,1right]$


  • $varphi$ is continuously differentiable on $mathbb{R}$ at every order.


  • $varphi$ is constant on (at least one) an intevarl of the form $left[0,1/nright]$, where $ninmathbb{N}$.


As pointed out in an exercise in the book "Functional Analysis, Sobolev Spaces, and Differential Equations" by Haim Brezis, we can construct one via mollifiers. However, it does not seem to give a closed form or explicit formula that we can easily play with.



So here is my question: Is there an example where we have a closed form that is easy to play with?



Any help/hint is highly appreciated.






real-analysis integration functional-analysis derivatives distribution-theory







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    This might be helpful.
    $endgroup$
    – MisterRiemann
    Nov 30 '18 at 18:33






  • 2




    $begingroup$
    You mean *mollifiers, not modifiers, presumably
    $endgroup$
    – qbert
    Nov 30 '18 at 18:37






  • 1




    $begingroup$
    I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
    $endgroup$
    – Clayton
    Nov 30 '18 at 18:38










  • $begingroup$
    @MisterRiemann Yes, very helpful indeed.
    $endgroup$
    – weirdo
    Nov 30 '18 at 18:38






  • 2




    $begingroup$
    Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
    $endgroup$
    – reuns
    Nov 30 '18 at 18:47
















1












1








1





$begingroup$


I am interested in finding a function $varphicolonmathbb{R}tomathbb{R}$ with the following property:





  • $varphi$ has compact support, which contains $left[0,1right]$


  • $varphi$ is continuously differentiable on $mathbb{R}$ at every order.


  • $varphi$ is constant on (at least one) an intevarl of the form $left[0,1/nright]$, where $ninmathbb{N}$.


As pointed out in an exercise in the book "Functional Analysis, Sobolev Spaces, and Differential Equations" by Haim Brezis, we can construct one via mollifiers. However, it does not seem to give a closed form or explicit formula that we can easily play with.



So here is my question: Is there an example where we have a closed form that is easy to play with?



Any help/hint is highly appreciated.






real-analysis integration functional-analysis derivatives distribution-theory







share|cite|improve this question











$endgroup$




I am interested in finding a function $varphicolonmathbb{R}tomathbb{R}$ with the following property:





  • $varphi$ has compact support, which contains $left[0,1right]$


  • $varphi$ is continuously differentiable on $mathbb{R}$ at every order.


  • $varphi$ is constant on (at least one) an intevarl of the form $left[0,1/nright]$, where $ninmathbb{N}$.


As pointed out in an exercise in the book "Functional Analysis, Sobolev Spaces, and Differential Equations" by Haim Brezis, we can construct one via mollifiers. However, it does not seem to give a closed form or explicit formula that we can easily play with.



So here is my question: Is there an example where we have a closed form that is easy to play with?



Any help/hint is highly appreciated.






real-analysis integration functional-analysis derivatives distribution-theory




real-analysis integration functional-analysis derivatives distribution-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 18:38







weirdo

















asked Nov 30 '18 at 18:31









weirdoweirdo

420210




420210








  • 1




    $begingroup$
    This might be helpful.
    $endgroup$
    – MisterRiemann
    Nov 30 '18 at 18:33






  • 2




    $begingroup$
    You mean *mollifiers, not modifiers, presumably
    $endgroup$
    – qbert
    Nov 30 '18 at 18:37






  • 1




    $begingroup$
    I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
    $endgroup$
    – Clayton
    Nov 30 '18 at 18:38










  • $begingroup$
    @MisterRiemann Yes, very helpful indeed.
    $endgroup$
    – weirdo
    Nov 30 '18 at 18:38






  • 2




    $begingroup$
    Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
    $endgroup$
    – reuns
    Nov 30 '18 at 18:47
















  • 1




    $begingroup$
    This might be helpful.
    $endgroup$
    – MisterRiemann
    Nov 30 '18 at 18:33






  • 2




    $begingroup$
    You mean *mollifiers, not modifiers, presumably
    $endgroup$
    – qbert
    Nov 30 '18 at 18:37






  • 1




    $begingroup$
    I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
    $endgroup$
    – Clayton
    Nov 30 '18 at 18:38










  • $begingroup$
    @MisterRiemann Yes, very helpful indeed.
    $endgroup$
    – weirdo
    Nov 30 '18 at 18:38






  • 2




    $begingroup$
    Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
    $endgroup$
    – reuns
    Nov 30 '18 at 18:47










1




1




$begingroup$
This might be helpful.
$endgroup$
– MisterRiemann
Nov 30 '18 at 18:33




$begingroup$
This might be helpful.
$endgroup$
– MisterRiemann
Nov 30 '18 at 18:33




2




2




$begingroup$
You mean *mollifiers, not modifiers, presumably
$endgroup$
– qbert
Nov 30 '18 at 18:37




$begingroup$
You mean *mollifiers, not modifiers, presumably
$endgroup$
– qbert
Nov 30 '18 at 18:37




1




1




$begingroup$
I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
$endgroup$
– Clayton
Nov 30 '18 at 18:38




$begingroup$
I think an easier way to state your second condition is simply to say $varphi$ is $C^infty$.
$endgroup$
– Clayton
Nov 30 '18 at 18:38












$begingroup$
@MisterRiemann Yes, very helpful indeed.
$endgroup$
– weirdo
Nov 30 '18 at 18:38




$begingroup$
@MisterRiemann Yes, very helpful indeed.
$endgroup$
– weirdo
Nov 30 '18 at 18:38




2




2




$begingroup$
Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
$endgroup$
– reuns
Nov 30 '18 at 18:47






$begingroup$
Same as MisterRiemann example : show one by one $C^infty$, compact support, constantness
$endgroup$
– reuns
Nov 30 '18 at 18:47












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