Integrability of composite functions [duplicate]
$begingroup$
This question already has an answer here:
Composition of two Riemann integrable functions
1 answer
Let $f$ be a Riemann-integrable function on a closed interval $[a,b] subset mathbb{R}$. Let g be a function on $mathbb{R}$. What conditions must g satisfy so that $g circ f$ is also Riemann-integrable ? Thank you!
real-analysis calculus integration definite-integrals riemann-integration
asked Dec 7 '18 at 5:32
user518704user518704
336
$endgroup$
marked as duplicate by KReiser, Brahadeesh, José Carlos Santos calculus
Users with the calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 7 '18 at 8:38
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:
Composition of two Riemann integrable functions
1 answer
Let $f$ be a Riemann-integrable function on a closed interval $[a,b] subset mathbb{R}$. Let g be a function on $mathbb{R}$. What conditions must g satisfy so that $g circ f$ is also Riemann-integrable ? Thank you!
real-analysis calculus integration definite-integrals riemann-integration
asked Dec 7 '18 at 5:32
user518704user518704
336
$endgroup$
marked as duplicate by KReiser, Brahadeesh, José Carlos Santos calculus
Users with the calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 7 '18 at 8:38
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.
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33
add a comment |
$begingroup$
This question already has an answer here:
Composition of two Riemann integrable functions
1 answer
Let $f$ be a Riemann-integrable function on a closed interval $[a,b] subset mathbb{R}$. Let g be a function on $mathbb{R}$. What conditions must g satisfy so that $g circ f$ is also Riemann-integrable ? Thank you!
real-analysis calculus integration definite-integrals riemann-integration
asked Dec 7 '18 at 5:32
user518704user518704
336
$endgroup$
This question already has an answer here:
Composition of two Riemann integrable functions
1 answer
Let $f$ be a Riemann-integrable function on a closed interval $[a,b] subset mathbb{R}$. Let g be a function on $mathbb{R}$. What conditions must g satisfy so that $g circ f$ is also Riemann-integrable ? Thank you!
This question already has an answer here:
Composition of two Riemann integrable functions
1 answer
real-analysis calculus integration definite-integrals riemann-integration
real-analysis calculus integration definite-integrals riemann-integration
asked Dec 7 '18 at 5:32
user518704user518704
336
asked Dec 7 '18 at 5:32
user518704user518704
336
asked Dec 7 '18 at 5:32
user518704user518704
336
asked Dec 7 '18 at 5:32
user518704user518704
336
asked Dec 7 '18 at 5:32
user518704user518704
336
336
marked as duplicate by KReiser, Brahadeesh, José Carlos Santos calculus
Users with the calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 7 '18 at 8:38
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 KReiser, Brahadeesh, José Carlos Santos calculus
Users with the calculus badge can single-handedly close calculus questions as duplicates and reopen them as needed.
StackExchange.ready(function() {
if (StackExchange.options.isMobile) return;
$('.dupe-hammer-message-hover:not(.hover-bound)').each(function() {
var $hover = $(this).addClass('hover-bound'),
$msg = $hover.siblings('.dupe-hammer-message');
$hover.hover(
function() {
$hover.showInfoMessage('', {
messageElement: $msg.clone().show(),
transient: false,
position: { my: 'bottom left', at: 'top center', offsetTop: -7 },
dismissable: false,
relativeToBody: true
});
},
function() {
StackExchange.helpers.removeMessages();
}
);
});
});
Dec 7 '18 at 8:38
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.
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33
add a comment |
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
See this post Riemann Integrability of Compositions
If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.
Quote from Is the composite function integrable? See the last result mentioned in the paper.
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
See this post Riemann Integrability of Compositions
If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.
Quote from Is the composite function integrable? See the last result mentioned in the paper.
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
$endgroup$
add a comment |
$begingroup$
See this post Riemann Integrability of Compositions
If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.
Quote from Is the composite function integrable? See the last result mentioned in the paper.
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
$endgroup$
add a comment |
$begingroup$
See this post Riemann Integrability of Compositions
If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.
Quote from Is the composite function integrable? See the last result mentioned in the paper.
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
$endgroup$
See this post Riemann Integrability of Compositions
If $f$ is a Riemann integrable function defined on $[a,b], g$ is a differentiable function with non-zero continuous derivative on $[c,d]$ and the range of $g$ is contained in $[a,b]$, then $fcirc g$ is Riemann integrable on $[c,d]$.
Quote from Is the composite function integrable? See the last result mentioned in the paper.
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
answered Dec 7 '18 at 5:56
William SunWilliam Sun
471111
471111
add a comment |
add a comment |
$begingroup$
@WilliamSun The answer to the linked question shows that if $g$ is Riemann-integrable then it is not necessary that $g circ f$ is Riemann-integrable. The question here is different, so it is not a duplicate. I am voting to close it as off-topic, though, because of the lack of context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:24
$begingroup$
@Brahadeesh Originally I planed to put the post mathoverflow.net/questions/20045/… in my answer below as a duplicate of the question. But the link I gave in my comment mentioned the post below in the comment section, which I think will provide the op with other useful informations. Another problem is I attempted to change the link of duplicate but I haven’t figure out how to do it.
$endgroup$
– William Sun
Dec 7 '18 at 8:30
$begingroup$
@WilliamSun I don't think it's possible to change the link of duplicate once you've voted. You've written a good answer, and I've upvoted it. But sadly this post will probably be closed as off-topic unless the OP improves it by providing more context.
$endgroup$
– Brahadeesh
Dec 7 '18 at 8:33