Composite of non Riemann integrable functions can be Riemann integrable?
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(1) Let $f,g$ be not Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we find an example such that $gcirc f(x)=g(f(x))$ is Riemann integrable on $[a,b]$?
(2) Let $f,g$ be Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we show that $gcirc f$ is Riemann integrable on $[a,b]$ also?
calculus integration
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
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add a comment |
$begingroup$
(1) Let $f,g$ be not Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we find an example such that $gcirc f(x)=g(f(x))$ is Riemann integrable on $[a,b]$?
(2) Let $f,g$ be Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we show that $gcirc f$ is Riemann integrable on $[a,b]$ also?
calculus integration
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
$endgroup$
$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08
add a comment |
$begingroup$
(1) Let $f,g$ be not Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we find an example such that $gcirc f(x)=g(f(x))$ is Riemann integrable on $[a,b]$?
(2) Let $f,g$ be Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we show that $gcirc f$ is Riemann integrable on $[a,b]$ also?
calculus integration
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
$endgroup$
(1) Let $f,g$ be not Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we find an example such that $gcirc f(x)=g(f(x))$ is Riemann integrable on $[a,b]$?
(2) Let $f,g$ be Riemann integrable on $[a,b]$, and the range of $f$ is $[a,b]$ also. Can we show that $gcirc f$ is Riemann integrable on $[a,b]$ also?
calculus integration
calculus integration
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
asked Dec 8 '18 at 5:00
xlddxldd
1,330510
1,330510
$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08
add a comment |
$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08
$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08
$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For question 1, $$f(x)=g(x)=
left{
begin{array}{ll}
x, &xin mathbb{Q} \
1-x,&xnotin mathbb{Q} \
end{array}
right.
\$$and $[a,b]=[0,1]$.
For question 2, see here.
In order to change the range of $f$, alter it to
$$ f(x) = begin{cases} 1/q & text{ for }x=p/qtext{ and $0le xle1/2$} \ 0 & text{ for } x notin mathbb{Q}text{ and $0le xle1/2$}\
2x-1 & text{others} end{cases}
$$ and the $g$ desired is the $f$ in my link.
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
$endgroup$
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
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@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
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@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
|
show 2 more comments
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
For question 1, $$f(x)=g(x)=
left{
begin{array}{ll}
x, &xin mathbb{Q} \
1-x,&xnotin mathbb{Q} \
end{array}
right.
\$$and $[a,b]=[0,1]$.
For question 2, see here.
In order to change the range of $f$, alter it to
$$ f(x) = begin{cases} 1/q & text{ for }x=p/qtext{ and $0le xle1/2$} \ 0 & text{ for } x notin mathbb{Q}text{ and $0le xle1/2$}\
2x-1 & text{others} end{cases}
$$ and the $g$ desired is the $f$ in my link.
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
$endgroup$
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
|
show 2 more comments
$begingroup$
For question 1, $$f(x)=g(x)=
left{
begin{array}{ll}
x, &xin mathbb{Q} \
1-x,&xnotin mathbb{Q} \
end{array}
right.
\$$and $[a,b]=[0,1]$.
For question 2, see here.
In order to change the range of $f$, alter it to
$$ f(x) = begin{cases} 1/q & text{ for }x=p/qtext{ and $0le xle1/2$} \ 0 & text{ for } x notin mathbb{Q}text{ and $0le xle1/2$}\
2x-1 & text{others} end{cases}
$$ and the $g$ desired is the $f$ in my link.
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
$endgroup$
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
|
show 2 more comments
$begingroup$
For question 1, $$f(x)=g(x)=
left{
begin{array}{ll}
x, &xin mathbb{Q} \
1-x,&xnotin mathbb{Q} \
end{array}
right.
\$$and $[a,b]=[0,1]$.
For question 2, see here.
In order to change the range of $f$, alter it to
$$ f(x) = begin{cases} 1/q & text{ for }x=p/qtext{ and $0le xle1/2$} \ 0 & text{ for } x notin mathbb{Q}text{ and $0le xle1/2$}\
2x-1 & text{others} end{cases}
$$ and the $g$ desired is the $f$ in my link.
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
$endgroup$
For question 1, $$f(x)=g(x)=
left{
begin{array}{ll}
x, &xin mathbb{Q} \
1-x,&xnotin mathbb{Q} \
end{array}
right.
\$$and $[a,b]=[0,1]$.
For question 2, see here.
In order to change the range of $f$, alter it to
$$ f(x) = begin{cases} 1/q & text{ for }x=p/qtext{ and $0le xle1/2$} \ 0 & text{ for } x notin mathbb{Q}text{ and $0le xle1/2$}\
2x-1 & text{others} end{cases}
$$ and the $g$ desired is the $f$ in my link.
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
edited Dec 10 '18 at 2:44
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
answered Dec 8 '18 at 5:25
Kemono ChenKemono Chen
3,1941844
3,1941844
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
|
show 2 more comments
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
question 2, there is another condition: the range of $f$ is $[a,b]$.
$endgroup$
– xldd
Dec 8 '18 at 7:29
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@xldd edited. Please check.
$endgroup$
– Kemono Chen
Dec 8 '18 at 7:53
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
@Chen What is the outer function and the inner function? It really troubles me.
$endgroup$
– xldd
Dec 8 '18 at 10:28
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
$begingroup$
In math.stackexchange.com/questions/1060834/…, the inner function is the Riemann function, whose range is not all of $[0,1]$, and the $f$ is the outer function, which seems you have changed to be here.
$endgroup$
– xldd
Dec 8 '18 at 10:30
1
1
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
$begingroup$
It is not hard to alter the inner function. Try to do it yourself. Anyway I edited it for correctness.
$endgroup$
– Kemono Chen
Dec 10 '18 at 2:45
|
show 2 more comments
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$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08