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The function defined by the sum of non-decreasing absolutely continuous functions on $mathbb{R}$ is...
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True or False? If ${f_n}_{nge 0}$ is a sequence of non-decreasing absolutely continuous functions on $mathbb{R}$, and if $f:=sum_{nge0}f_n(x), xinmathbb{R},$ is finite at each $x$, then $f$ is continuous and differentiable a.e. on $mathbb{R}$.
I was trying to show that $f$ is absolutely continuous on $mathbb{R}$ so that it would be continuous and differentiable a.e. But I failed to write it as an indefinite integral. Maybe there's some other way to prove it. Also, most of the theorems I learn concern the matter (differentiable a.e., or absolutely continuous etc.) on intervals $(a,b)$ or $[a,b]$, not on the entire $mathbb{R}$. Can anyone give me some hints?
real-analysis analysis continuity indefinite-integrals absolute-continuity
asked Jan 7 at 22:49
AlexAlex
567
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add a comment |
$begingroup$
True or False? If ${f_n}_{nge 0}$ is a sequence of non-decreasing absolutely continuous functions on $mathbb{R}$, and if $f:=sum_{nge0}f_n(x), xinmathbb{R},$ is finite at each $x$, then $f$ is continuous and differentiable a.e. on $mathbb{R}$.
I was trying to show that $f$ is absolutely continuous on $mathbb{R}$ so that it would be continuous and differentiable a.e. But I failed to write it as an indefinite integral. Maybe there's some other way to prove it. Also, most of the theorems I learn concern the matter (differentiable a.e., or absolutely continuous etc.) on intervals $(a,b)$ or $[a,b]$, not on the entire $mathbb{R}$. Can anyone give me some hints?
real-analysis analysis continuity indefinite-integrals absolute-continuity
asked Jan 7 at 22:49
AlexAlex
567
$endgroup$
$begingroup$
You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04
add a comment |
$begingroup$
True or False? If ${f_n}_{nge 0}$ is a sequence of non-decreasing absolutely continuous functions on $mathbb{R}$, and if $f:=sum_{nge0}f_n(x), xinmathbb{R},$ is finite at each $x$, then $f$ is continuous and differentiable a.e. on $mathbb{R}$.
I was trying to show that $f$ is absolutely continuous on $mathbb{R}$ so that it would be continuous and differentiable a.e. But I failed to write it as an indefinite integral. Maybe there's some other way to prove it. Also, most of the theorems I learn concern the matter (differentiable a.e., or absolutely continuous etc.) on intervals $(a,b)$ or $[a,b]$, not on the entire $mathbb{R}$. Can anyone give me some hints?
real-analysis analysis continuity indefinite-integrals absolute-continuity
asked Jan 7 at 22:49
AlexAlex
567
$endgroup$
True or False? If ${f_n}_{nge 0}$ is a sequence of non-decreasing absolutely continuous functions on $mathbb{R}$, and if $f:=sum_{nge0}f_n(x), xinmathbb{R},$ is finite at each $x$, then $f$ is continuous and differentiable a.e. on $mathbb{R}$.
I was trying to show that $f$ is absolutely continuous on $mathbb{R}$ so that it would be continuous and differentiable a.e. But I failed to write it as an indefinite integral. Maybe there's some other way to prove it. Also, most of the theorems I learn concern the matter (differentiable a.e., or absolutely continuous etc.) on intervals $(a,b)$ or $[a,b]$, not on the entire $mathbb{R}$. Can anyone give me some hints?
real-analysis analysis continuity indefinite-integrals absolute-continuity
real-analysis analysis continuity indefinite-integrals absolute-continuity
asked Jan 7 at 22:49
AlexAlex
567
asked Jan 7 at 22:49
AlexAlex
567
asked Jan 7 at 22:49
AlexAlex
567
asked Jan 7 at 22:49
AlexAlex
567
asked Jan 7 at 22:49
AlexAlex
567
567
$begingroup$
You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04
add a comment |
$begingroup$
You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04
$begingroup$
You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04
$begingroup$
You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04
add a comment |
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You can prove that the function is continuous on any interval, thus proving it is continuous; you can prove that the non-differentiability set has null measure in any interval, so it must have null measure in the whole line.
$endgroup$
– Mindlack
Jan 7 at 23:04