Prove that this function is not integrable: $f(x)$ is $1-x$ for $x$ rational, and $1/x$ for $x$ irrational...
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This question already has an answer here:
Proving a function is not Riemann integrable
1 answer
How do I prove that the function
$$f(x)= begin{cases}
1-x &,text{when}; x ;text{is rational} \
1/x &, text{when}; x ;text{is irrational}
end{cases}$$ is not integrable on every interval $[a,b]$?
real-analysis
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
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marked as duplicate by Anurag A, Lord_Farin, Xander Henderson, RRL, Shailesh Dec 17 '18 at 0:07
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Proving a function is not Riemann integrable
1 answer
How do I prove that the function
$$f(x)= begin{cases}
1-x &,text{when}; x ;text{is rational} \
1/x &, text{when}; x ;text{is irrational}
end{cases}$$ is not integrable on every interval $[a,b]$?
real-analysis
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
$endgroup$
marked as duplicate by Anurag A, Lord_Farin, Xander Henderson, RRL, Shailesh Dec 17 '18 at 0:07
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
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Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
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– Blue
Dec 16 '18 at 20:21
1
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Riemann or Lebesgue integral?
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– Brian Tung
Dec 16 '18 at 21:17
add a comment |
$begingroup$
This question already has an answer here:
Proving a function is not Riemann integrable
1 answer
How do I prove that the function
$$f(x)= begin{cases}
1-x &,text{when}; x ;text{is rational} \
1/x &, text{when}; x ;text{is irrational}
end{cases}$$ is not integrable on every interval $[a,b]$?
real-analysis
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
$endgroup$
This question already has an answer here:
Proving a function is not Riemann integrable
1 answer
How do I prove that the function
$$f(x)= begin{cases}
1-x &,text{when}; x ;text{is rational} \
1/x &, text{when}; x ;text{is irrational}
end{cases}$$ is not integrable on every interval $[a,b]$?
This question already has an answer here:
Proving a function is not Riemann integrable
1 answer
real-analysis
real-analysis
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
edited Dec 16 '18 at 20:56
the_candyman
8,94132145
8,94132145
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
asked Dec 16 '18 at 20:10
Băeșu Rareș GabrielBăeșu Rareș Gabriel
6
6
marked as duplicate by Anurag A, Lord_Farin, Xander Henderson, RRL, Shailesh Dec 17 '18 at 0:07
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Anurag A, Lord_Farin, Xander Henderson, RRL, Shailesh Dec 17 '18 at 0:07
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
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Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
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– Blue
Dec 16 '18 at 20:21
1
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Riemann or Lebesgue integral?
$endgroup$
– Brian Tung
Dec 16 '18 at 21:17
add a comment |
3
$begingroup$
Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
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– Blue
Dec 16 '18 at 20:21
1
$begingroup$
Riemann or Lebesgue integral?
$endgroup$
– Brian Tung
Dec 16 '18 at 21:17
3
3
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Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
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– Blue
Dec 16 '18 at 20:21
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Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
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– Blue
Dec 16 '18 at 20:21
1
1
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Riemann or Lebesgue integral?
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– Brian Tung
Dec 16 '18 at 21:17
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Riemann or Lebesgue integral?
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– Brian Tung
Dec 16 '18 at 21:17
add a comment |
1 Answer
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By the definition of integral, it is easy to check that for any partition of [a,b], the upper integral never equals the lower integral. Thus the function not integrable.
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
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1 Answer
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active
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1 Answer
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active
oldest
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active
oldest
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active
oldest
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By the definition of integral, it is easy to check that for any partition of [a,b], the upper integral never equals the lower integral. Thus the function not integrable.
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
$endgroup$
add a comment |
$begingroup$
By the definition of integral, it is easy to check that for any partition of [a,b], the upper integral never equals the lower integral. Thus the function not integrable.
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
$endgroup$
add a comment |
$begingroup$
By the definition of integral, it is easy to check that for any partition of [a,b], the upper integral never equals the lower integral. Thus the function not integrable.
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
$endgroup$
By the definition of integral, it is easy to check that for any partition of [a,b], the upper integral never equals the lower integral. Thus the function not integrable.
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
answered Dec 16 '18 at 20:36
Alvis NordkovichAlvis Nordkovich
1469
1469
add a comment |
add a comment |
3
$begingroup$
Welcome to Math.SE! You'll find that the community here prefers/expects a question to involve more than a simple problem statement. Please include something about what you know of the problem and/or where you got stuck. This information helps answerers tailor their responses to your skill level, while avoiding wasting time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Help us help you.
$endgroup$
– Blue
Dec 16 '18 at 20:21
1
$begingroup$
Riemann or Lebesgue integral?
$endgroup$
– Brian Tung
Dec 16 '18 at 21:17