Prove the continuous function is not necessarily absolutely continuous











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Let $f$ be continuous on $I=[0,1]$, and absolutly continuous on $[epsilon,1]$ for any $0<epsilon<1$.



(i) Show that $f$ may not be absolutely continuous on $I.$



(ii) Show that $f$ is absolutely continuous on $I$ if it is increasing.



(ii)Show that $f(x)=sqrt{x}$ is absolutly continuous BUT not Lipschitz on $I$.



I am struggling with the first part, I tried to find a finite disjoint collection of open intervals in $I$ such that $sum_{i=1}^nl(I_i)<delta$ for any $delta>0$, but $$sum_{i=1}^n|f(b_i)-f(a_i)|geq epsilon$$
where $I_i=(a_i,b_i).$ So I would appreciate any help with that.










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  • 3




    Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
    – Mark Viola
    8 hours ago












  • Can you explain that more please.
    – Ahmed
    7 hours ago















up vote
1
down vote

favorite












Let $f$ be continuous on $I=[0,1]$, and absolutly continuous on $[epsilon,1]$ for any $0<epsilon<1$.



(i) Show that $f$ may not be absolutely continuous on $I.$



(ii) Show that $f$ is absolutely continuous on $I$ if it is increasing.



(ii)Show that $f(x)=sqrt{x}$ is absolutly continuous BUT not Lipschitz on $I$.



I am struggling with the first part, I tried to find a finite disjoint collection of open intervals in $I$ such that $sum_{i=1}^nl(I_i)<delta$ for any $delta>0$, but $$sum_{i=1}^n|f(b_i)-f(a_i)|geq epsilon$$
where $I_i=(a_i,b_i).$ So I would appreciate any help with that.










share|cite|improve this question


















  • 3




    Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
    – Mark Viola
    8 hours ago












  • Can you explain that more please.
    – Ahmed
    7 hours ago













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $f$ be continuous on $I=[0,1]$, and absolutly continuous on $[epsilon,1]$ for any $0<epsilon<1$.



(i) Show that $f$ may not be absolutely continuous on $I.$



(ii) Show that $f$ is absolutely continuous on $I$ if it is increasing.



(ii)Show that $f(x)=sqrt{x}$ is absolutly continuous BUT not Lipschitz on $I$.



I am struggling with the first part, I tried to find a finite disjoint collection of open intervals in $I$ such that $sum_{i=1}^nl(I_i)<delta$ for any $delta>0$, but $$sum_{i=1}^n|f(b_i)-f(a_i)|geq epsilon$$
where $I_i=(a_i,b_i).$ So I would appreciate any help with that.










share|cite|improve this question













Let $f$ be continuous on $I=[0,1]$, and absolutly continuous on $[epsilon,1]$ for any $0<epsilon<1$.



(i) Show that $f$ may not be absolutely continuous on $I.$



(ii) Show that $f$ is absolutely continuous on $I$ if it is increasing.



(ii)Show that $f(x)=sqrt{x}$ is absolutly continuous BUT not Lipschitz on $I$.



I am struggling with the first part, I tried to find a finite disjoint collection of open intervals in $I$ such that $sum_{i=1}^nl(I_i)<delta$ for any $delta>0$, but $$sum_{i=1}^n|f(b_i)-f(a_i)|geq epsilon$$
where $I_i=(a_i,b_i).$ So I would appreciate any help with that.







measure-theory lebesgue-measure






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share|cite|improve this question











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share|cite|improve this question










asked 8 hours ago









Ahmed

26819




26819








  • 3




    Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
    – Mark Viola
    8 hours ago












  • Can you explain that more please.
    – Ahmed
    7 hours ago














  • 3




    Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
    – Mark Viola
    8 hours ago












  • Can you explain that more please.
    – Ahmed
    7 hours ago








3




3




Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
– Mark Viola
8 hours ago






Let $f(x) = xsin(x) $ for $xne 0$ and $f(0)=0$. $f$ is continuous but not absolutely continuous on any interval that contains $0$.
– Mark Viola
8 hours ago














Can you explain that more please.
– Ahmed
7 hours ago




Can you explain that more please.
– Ahmed
7 hours ago










1 Answer
1






active

oldest

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up vote
0
down vote













HINT:



Let $f$ be the function given by



$$f(x)=begin{cases}
xsin(pi/2x)&,xne0\\
0&,x=0
end{cases}$$



Take $epsilon=1$. Let $delta >0$ be given.



Then, take $x_k =frac1{Nk}$ and $y_k=frac1{N(k+1)}$ for $N$ and odd integer and $1/delta <N$.



Show that the sum $sum_{k=1}^N|x_k-y_k|<delta$, but $sum_{k=1}^N|f(x_k)-f(y_k) |ge1$.






share|cite|improve this answer























  • @ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
    – Mark Viola
    4 hours ago










  • And please feel free to up vote and accept an answer as you see fit of course. ;-)
    – Mark Viola
    4 hours ago











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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
0
down vote













HINT:



Let $f$ be the function given by



$$f(x)=begin{cases}
xsin(pi/2x)&,xne0\\
0&,x=0
end{cases}$$



Take $epsilon=1$. Let $delta >0$ be given.



Then, take $x_k =frac1{Nk}$ and $y_k=frac1{N(k+1)}$ for $N$ and odd integer and $1/delta <N$.



Show that the sum $sum_{k=1}^N|x_k-y_k|<delta$, but $sum_{k=1}^N|f(x_k)-f(y_k) |ge1$.






share|cite|improve this answer























  • @ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
    – Mark Viola
    4 hours ago










  • And please feel free to up vote and accept an answer as you see fit of course. ;-)
    – Mark Viola
    4 hours ago















up vote
0
down vote













HINT:



Let $f$ be the function given by



$$f(x)=begin{cases}
xsin(pi/2x)&,xne0\\
0&,x=0
end{cases}$$



Take $epsilon=1$. Let $delta >0$ be given.



Then, take $x_k =frac1{Nk}$ and $y_k=frac1{N(k+1)}$ for $N$ and odd integer and $1/delta <N$.



Show that the sum $sum_{k=1}^N|x_k-y_k|<delta$, but $sum_{k=1}^N|f(x_k)-f(y_k) |ge1$.






share|cite|improve this answer























  • @ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
    – Mark Viola
    4 hours ago










  • And please feel free to up vote and accept an answer as you see fit of course. ;-)
    – Mark Viola
    4 hours ago













up vote
0
down vote










up vote
0
down vote









HINT:



Let $f$ be the function given by



$$f(x)=begin{cases}
xsin(pi/2x)&,xne0\\
0&,x=0
end{cases}$$



Take $epsilon=1$. Let $delta >0$ be given.



Then, take $x_k =frac1{Nk}$ and $y_k=frac1{N(k+1)}$ for $N$ and odd integer and $1/delta <N$.



Show that the sum $sum_{k=1}^N|x_k-y_k|<delta$, but $sum_{k=1}^N|f(x_k)-f(y_k) |ge1$.






share|cite|improve this answer














HINT:



Let $f$ be the function given by



$$f(x)=begin{cases}
xsin(pi/2x)&,xne0\\
0&,x=0
end{cases}$$



Take $epsilon=1$. Let $delta >0$ be given.



Then, take $x_k =frac1{Nk}$ and $y_k=frac1{N(k+1)}$ for $N$ and odd integer and $1/delta <N$.



Show that the sum $sum_{k=1}^N|x_k-y_k|<delta$, but $sum_{k=1}^N|f(x_k)-f(y_k) |ge1$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 6 hours ago

























answered 6 hours ago









Mark Viola

129k1273170




129k1273170












  • @ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
    – Mark Viola
    4 hours ago










  • And please feel free to up vote and accept an answer as you see fit of course. ;-)
    – Mark Viola
    4 hours ago


















  • @ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
    – Mark Viola
    4 hours ago










  • And please feel free to up vote and accept an answer as you see fit of course. ;-)
    – Mark Viola
    4 hours ago
















@ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
– Mark Viola
4 hours ago




@ahmed Please let me know how I can improve my answer. I really want to give you the best answer I can.
– Mark Viola
4 hours ago












And please feel free to up vote and accept an answer as you see fit of course. ;-)
– Mark Viola
4 hours ago




And please feel free to up vote and accept an answer as you see fit of course. ;-)
– Mark Viola
4 hours ago


















 

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