Example of $f,g: [0,1]to[0,1]$ and Riemann-integrable, but $gcirc f$ is not?












4












$begingroup$


Give me an example of two Riemann-integrable functions $f,g:[0,1]to[0,1]$ such that $gcirc f$ isn't integrable!
I already know the following example:
$$f(x)=begin{cases}
0, & text{if $x$ is irrational} \
1, & text{if $x=0$}\
frac1q, & text{if $x$ is rational and $x=frac pq$ such that $qinBbb N$ and $(p,q)=1$} \
end{cases}$$
$$g(x) =
begin{cases}
1, & text{if $x$ is of the form $frac 1q$such that $qin Bbb N$} \
0, & text{otherwise} \
end{cases}$$
now observe that $gcirc f$ is a famous example of non-integrable function!






real-analysis calculus integration examples-counterexamples riemann-integration







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
    $endgroup$
    – 6005
    Mar 7 '14 at 20:30










  • $begingroup$
    Extend to what??
    $endgroup$
    – k1.M
    Mar 7 '14 at 21:12










  • $begingroup$
    Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
    $endgroup$
    – 6005
    Mar 8 '14 at 9:10










  • $begingroup$
    Can someone explain why f is integrable?
    $endgroup$
    – Zslice
    Dec 17 '14 at 10:32
















4












$begingroup$


Give me an example of two Riemann-integrable functions $f,g:[0,1]to[0,1]$ such that $gcirc f$ isn't integrable!
I already know the following example:
$$f(x)=begin{cases}
0, & text{if $x$ is irrational} \
1, & text{if $x=0$}\
frac1q, & text{if $x$ is rational and $x=frac pq$ such that $qinBbb N$ and $(p,q)=1$} \
end{cases}$$
$$g(x) =
begin{cases}
1, & text{if $x$ is of the form $frac 1q$such that $qin Bbb N$} \
0, & text{otherwise} \
end{cases}$$
now observe that $gcirc f$ is a famous example of non-integrable function!






real-analysis calculus integration examples-counterexamples riemann-integration







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
    $endgroup$
    – 6005
    Mar 7 '14 at 20:30










  • $begingroup$
    Extend to what??
    $endgroup$
    – k1.M
    Mar 7 '14 at 21:12










  • $begingroup$
    Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
    $endgroup$
    – 6005
    Mar 8 '14 at 9:10










  • $begingroup$
    Can someone explain why f is integrable?
    $endgroup$
    – Zslice
    Dec 17 '14 at 10:32














4












4








4


3



$begingroup$


Give me an example of two Riemann-integrable functions $f,g:[0,1]to[0,1]$ such that $gcirc f$ isn't integrable!
I already know the following example:
$$f(x)=begin{cases}
0, & text{if $x$ is irrational} \
1, & text{if $x=0$}\
frac1q, & text{if $x$ is rational and $x=frac pq$ such that $qinBbb N$ and $(p,q)=1$} \
end{cases}$$
$$g(x) =
begin{cases}
1, & text{if $x$ is of the form $frac 1q$such that $qin Bbb N$} \
0, & text{otherwise} \
end{cases}$$
now observe that $gcirc f$ is a famous example of non-integrable function!






real-analysis calculus integration examples-counterexamples riemann-integration







share|cite|improve this question











$endgroup$




Give me an example of two Riemann-integrable functions $f,g:[0,1]to[0,1]$ such that $gcirc f$ isn't integrable!
I already know the following example:
$$f(x)=begin{cases}
0, & text{if $x$ is irrational} \
1, & text{if $x=0$}\
frac1q, & text{if $x$ is rational and $x=frac pq$ such that $qinBbb N$ and $(p,q)=1$} \
end{cases}$$
$$g(x) =
begin{cases}
1, & text{if $x$ is of the form $frac 1q$such that $qin Bbb N$} \
0, & text{otherwise} \
end{cases}$$
now observe that $gcirc f$ is a famous example of non-integrable function!






real-analysis calculus integration examples-counterexamples riemann-integration




real-analysis calculus integration examples-counterexamples riemann-integration






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 19 at 4:00









Martin Sleziak

45.1k10123277




45.1k10123277










asked Mar 7 '14 at 20:28









k1.Mk1.M

4,0711137




4,0711137








  • 2




    $begingroup$
    As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
    $endgroup$
    – 6005
    Mar 7 '14 at 20:30










  • $begingroup$
    Extend to what??
    $endgroup$
    – k1.M
    Mar 7 '14 at 21:12










  • $begingroup$
    Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
    $endgroup$
    – 6005
    Mar 8 '14 at 9:10










  • $begingroup$
    Can someone explain why f is integrable?
    $endgroup$
    – Zslice
    Dec 17 '14 at 10:32














  • 2




    $begingroup$
    As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
    $endgroup$
    – 6005
    Mar 7 '14 at 20:30










  • $begingroup$
    Extend to what??
    $endgroup$
    – k1.M
    Mar 7 '14 at 21:12










  • $begingroup$
    Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
    $endgroup$
    – 6005
    Mar 8 '14 at 9:10










  • $begingroup$
    Can someone explain why f is integrable?
    $endgroup$
    – Zslice
    Dec 17 '14 at 10:32








2




2




$begingroup$
As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
$endgroup$
– 6005
Mar 7 '14 at 20:30




$begingroup$
As it turns out, this particular $f$ and $g$ are not a great example--when you extend the definition of the integral to include more functions, $g circ f$ integrates fine.
$endgroup$
– 6005
Mar 7 '14 at 20:30












$begingroup$
Extend to what??
$endgroup$
– k1.M
Mar 7 '14 at 21:12




$begingroup$
Extend to what??
$endgroup$
– k1.M
Mar 7 '14 at 21:12












$begingroup$
Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
$endgroup$
– 6005
Mar 8 '14 at 9:10




$begingroup$
Lebesgue integration! The integral of $g circ f$ turns out to be $0$.
$endgroup$
– 6005
Mar 8 '14 at 9:10












$begingroup$
Can someone explain why f is integrable?
$endgroup$
– Zslice
Dec 17 '14 at 10:32




$begingroup$
Can someone explain why f is integrable?
$endgroup$
– Zslice
Dec 17 '14 at 10:32










1 Answer
1






active

oldest

votes


















5












$begingroup$

More simply, sticking with the same $f$ you may consider
$$g(x) =
begin{cases}
0, & text{if $x=0$} \
1, & text{if $xin ]0,1]$} \
end{cases}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    in this case$gcirc f$ is integrable!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:49








  • 4




    $begingroup$
    Nope it is not ! I'll provide proof if you're not persuaded.
    $endgroup$
    – Gabriel Romon
    Mar 7 '14 at 20:54










  • $begingroup$
    excuse me your answer is true and simpler thanks!!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:57






  • 3




    $begingroup$
    You seem to like exclamation points!!!!!
    $endgroup$
    – recursive recursion
    Mar 7 '14 at 21:26












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

More simply, sticking with the same $f$ you may consider
$$g(x) =
begin{cases}
0, & text{if $x=0$} \
1, & text{if $xin ]0,1]$} \
end{cases}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    in this case$gcirc f$ is integrable!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:49








  • 4




    $begingroup$
    Nope it is not ! I'll provide proof if you're not persuaded.
    $endgroup$
    – Gabriel Romon
    Mar 7 '14 at 20:54










  • $begingroup$
    excuse me your answer is true and simpler thanks!!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:57






  • 3




    $begingroup$
    You seem to like exclamation points!!!!!
    $endgroup$
    – recursive recursion
    Mar 7 '14 at 21:26
















5












$begingroup$

More simply, sticking with the same $f$ you may consider
$$g(x) =
begin{cases}
0, & text{if $x=0$} \
1, & text{if $xin ]0,1]$} \
end{cases}$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    in this case$gcirc f$ is integrable!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:49








  • 4




    $begingroup$
    Nope it is not ! I'll provide proof if you're not persuaded.
    $endgroup$
    – Gabriel Romon
    Mar 7 '14 at 20:54










  • $begingroup$
    excuse me your answer is true and simpler thanks!!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:57






  • 3




    $begingroup$
    You seem to like exclamation points!!!!!
    $endgroup$
    – recursive recursion
    Mar 7 '14 at 21:26














5












5








5





$begingroup$

More simply, sticking with the same $f$ you may consider
$$g(x) =
begin{cases}
0, & text{if $x=0$} \
1, & text{if $xin ]0,1]$} \
end{cases}$$






share|cite|improve this answer









$endgroup$



More simply, sticking with the same $f$ you may consider
$$g(x) =
begin{cases}
0, & text{if $x=0$} \
1, & text{if $xin ]0,1]$} \
end{cases}$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 7 '14 at 20:42









Gabriel RomonGabriel Romon

18.1k53387




18.1k53387












  • $begingroup$
    in this case$gcirc f$ is integrable!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:49








  • 4




    $begingroup$
    Nope it is not ! I'll provide proof if you're not persuaded.
    $endgroup$
    – Gabriel Romon
    Mar 7 '14 at 20:54










  • $begingroup$
    excuse me your answer is true and simpler thanks!!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:57






  • 3




    $begingroup$
    You seem to like exclamation points!!!!!
    $endgroup$
    – recursive recursion
    Mar 7 '14 at 21:26


















  • $begingroup$
    in this case$gcirc f$ is integrable!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:49








  • 4




    $begingroup$
    Nope it is not ! I'll provide proof if you're not persuaded.
    $endgroup$
    – Gabriel Romon
    Mar 7 '14 at 20:54










  • $begingroup$
    excuse me your answer is true and simpler thanks!!
    $endgroup$
    – k1.M
    Mar 7 '14 at 20:57






  • 3




    $begingroup$
    You seem to like exclamation points!!!!!
    $endgroup$
    – recursive recursion
    Mar 7 '14 at 21:26
















$begingroup$
in this case$gcirc f$ is integrable!
$endgroup$
– k1.M
Mar 7 '14 at 20:49






$begingroup$
in this case$gcirc f$ is integrable!
$endgroup$
– k1.M
Mar 7 '14 at 20:49






4




4




$begingroup$
Nope it is not ! I'll provide proof if you're not persuaded.
$endgroup$
– Gabriel Romon
Mar 7 '14 at 20:54




$begingroup$
Nope it is not ! I'll provide proof if you're not persuaded.
$endgroup$
– Gabriel Romon
Mar 7 '14 at 20:54












$begingroup$
excuse me your answer is true and simpler thanks!!
$endgroup$
– k1.M
Mar 7 '14 at 20:57




$begingroup$
excuse me your answer is true and simpler thanks!!
$endgroup$
– k1.M
Mar 7 '14 at 20:57




3




3




$begingroup$
You seem to like exclamation points!!!!!
$endgroup$
– recursive recursion
Mar 7 '14 at 21:26




$begingroup$
You seem to like exclamation points!!!!!
$endgroup$
– recursive recursion
Mar 7 '14 at 21:26


















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