Composite of non Riemann integrable functions can be Riemann integrable?












1












$begingroup$


(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







share|cite|improve this question









$endgroup$












  • $begingroup$
    Also, please avoid multiple questions in a post.
    $endgroup$
    – Kemono Chen
    Dec 8 '18 at 5:08


















1












$begingroup$


(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







share|cite|improve this question









$endgroup$












  • $begingroup$
    Also, please avoid multiple questions in a post.
    $endgroup$
    – Kemono Chen
    Dec 8 '18 at 5:08
















1












1








1





$begingroup$


(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







share|cite|improve this question









$endgroup$




(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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 8 '18 at 5:00









xlddxldd

1,315510




1,315510












  • $begingroup$
    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






$begingroup$
Also, please avoid multiple questions in a post.
$endgroup$
– Kemono Chen
Dec 8 '18 at 5:08












1 Answer
1






active

oldest

votes


















2












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030710%2fcomposite-of-non-riemann-integrable-functions-can-be-riemann-integrable%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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
















2












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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














2












2








2





$begingroup$

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.






share|cite|improve this answer











$endgroup$



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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 10 '18 at 2:44

























answered Dec 8 '18 at 5:25









Kemono ChenKemono Chen

3,0321743




3,0321743












  • $begingroup$
    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










  • $begingroup$
    @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


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030710%2fcomposite-of-non-riemann-integrable-functions-can-be-riemann-integrable%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Wiesbaden

Image
Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
Read more

Marschland

Image
Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
Read more

Dieringhausen

Image
Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
Read more