Riemann-Darboux Integrability of Subinterval












0














I'm studying Riemann-Darboux integration. I'm trying to prove the following rather intuitive notion for integrals. Please let me know if you find any errors in this proof, as I'm self-studying this topic.



Theorem: Suppose $f$ is Riemann-Darboux integrable on $[a,b]$. Let $cin(a,b)$. Then, $f$ is Riemann-Darboux integrable on the intervals $[a,c]$ and $[c,b]$.



Attempted Proof: Since $f$ is Riemann-Darboux integrable on $[a,b]$, for arbitrary $epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<epsilon$.



Let $n_p,n_{p*}$, and $n_{p'}$ be the number of partition parts in $P$, $P^*$, and $P'$, respectively. Also, let $m_i=inf_{xin[x_{i-1},x_i]}f(x)$ and $M_i=sup_{xin[x_{i-1},x_i]}f(x)$.



Consider the partition of $[a,c]$ given by $P^*=Pcap[a,c]$. Then, $$U(f,P^*)-L(f,P^*)=sum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[a,c]$.



Next, consider the partition $[a,b]$ given by $P'=Pcap[c,b]$. Then,



$$U(f,P')-L(f,P')=sum_{i=n_{p*}+1}^{n_{p'}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[c,b]$. $square$



Any and all feedback, or alternative proofs are appreciated. I love to see different arguments to expand my skill set.






real-analysis integration







share|cite|improve this question



























    0














    I'm studying Riemann-Darboux integration. I'm trying to prove the following rather intuitive notion for integrals. Please let me know if you find any errors in this proof, as I'm self-studying this topic.



    Theorem: Suppose $f$ is Riemann-Darboux integrable on $[a,b]$. Let $cin(a,b)$. Then, $f$ is Riemann-Darboux integrable on the intervals $[a,c]$ and $[c,b]$.



    Attempted Proof: Since $f$ is Riemann-Darboux integrable on $[a,b]$, for arbitrary $epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<epsilon$.



    Let $n_p,n_{p*}$, and $n_{p'}$ be the number of partition parts in $P$, $P^*$, and $P'$, respectively. Also, let $m_i=inf_{xin[x_{i-1},x_i]}f(x)$ and $M_i=sup_{xin[x_{i-1},x_i]}f(x)$.



    Consider the partition of $[a,c]$ given by $P^*=Pcap[a,c]$. Then, $$U(f,P^*)-L(f,P^*)=sum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[a,c]$.



    Next, consider the partition $[a,b]$ given by $P'=Pcap[c,b]$. Then,



    $$U(f,P')-L(f,P')=sum_{i=n_{p*}+1}^{n_{p'}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[c,b]$. $square$



    Any and all feedback, or alternative proofs are appreciated. I love to see different arguments to expand my skill set.






    real-analysis integration







    share|cite|improve this question

























      0












      0








      0







      I'm studying Riemann-Darboux integration. I'm trying to prove the following rather intuitive notion for integrals. Please let me know if you find any errors in this proof, as I'm self-studying this topic.



      Theorem: Suppose $f$ is Riemann-Darboux integrable on $[a,b]$. Let $cin(a,b)$. Then, $f$ is Riemann-Darboux integrable on the intervals $[a,c]$ and $[c,b]$.



      Attempted Proof: Since $f$ is Riemann-Darboux integrable on $[a,b]$, for arbitrary $epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<epsilon$.



      Let $n_p,n_{p*}$, and $n_{p'}$ be the number of partition parts in $P$, $P^*$, and $P'$, respectively. Also, let $m_i=inf_{xin[x_{i-1},x_i]}f(x)$ and $M_i=sup_{xin[x_{i-1},x_i]}f(x)$.



      Consider the partition of $[a,c]$ given by $P^*=Pcap[a,c]$. Then, $$U(f,P^*)-L(f,P^*)=sum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[a,c]$.



      Next, consider the partition $[a,b]$ given by $P'=Pcap[c,b]$. Then,



      $$U(f,P')-L(f,P')=sum_{i=n_{p*}+1}^{n_{p'}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[c,b]$. $square$



      Any and all feedback, or alternative proofs are appreciated. I love to see different arguments to expand my skill set.






      real-analysis integration







      share|cite|improve this question













      I'm studying Riemann-Darboux integration. I'm trying to prove the following rather intuitive notion for integrals. Please let me know if you find any errors in this proof, as I'm self-studying this topic.



      Theorem: Suppose $f$ is Riemann-Darboux integrable on $[a,b]$. Let $cin(a,b)$. Then, $f$ is Riemann-Darboux integrable on the intervals $[a,c]$ and $[c,b]$.



      Attempted Proof: Since $f$ is Riemann-Darboux integrable on $[a,b]$, for arbitrary $epsilon>0$, there exists a partition $P$ of $[a,b]$ such that $U(f,P)-L(f,P)<epsilon$.



      Let $n_p,n_{p*}$, and $n_{p'}$ be the number of partition parts in $P$, $P^*$, and $P'$, respectively. Also, let $m_i=inf_{xin[x_{i-1},x_i]}f(x)$ and $M_i=sup_{xin[x_{i-1},x_i]}f(x)$.



      Consider the partition of $[a,c]$ given by $P^*=Pcap[a,c]$. Then, $$U(f,P^*)-L(f,P^*)=sum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[a,c]$.



      Next, consider the partition $[a,b]$ given by $P'=Pcap[c,b]$. Then,



      $$U(f,P')-L(f,P')=sum_{i=n_{p*}+1}^{n_{p'}}(M_i-m_i)Delta x_ilesum_{i=1}^{n_{p*}}(M_i-m_i)Delta x_i+ sum_{n_{p*}+1}^{n_p}(M_i-m_i)Delta x_i=sum_{i=1}^{n_p}(M_i-m_i)Delta x_i=U(f,P)-L(f,P)<epsilon$$ Therefore, $f$ is integrable on $[c,b]$. $square$



      Any and all feedback, or alternative proofs are appreciated. I love to see different arguments to expand my skill set.






      real-analysis integration




      real-analysis integration






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 28 at 3:48









      Tree

      1046




      1046






















          1 Answer
          1






          active

          oldest

          votes


















          2














          This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that
          $$U(P,f)-L(P,f)<epsilon$$
          is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have
          $$U(P_{c},f)-L(P_{c},f)leq U(P,f)-L(P,f)<epsilon$$
          and you can proceed as you have done in your proof.



          An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.






          share|cite|improve this answer





















            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%2f3016686%2friemann-darboux-integrability-of-subinterval%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














            This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that
            $$U(P,f)-L(P,f)<epsilon$$
            is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have
            $$U(P_{c},f)-L(P_{c},f)leq U(P,f)-L(P,f)<epsilon$$
            and you can proceed as you have done in your proof.



            An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.






            share|cite|improve this answer


























              2














              This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that
              $$U(P,f)-L(P,f)<epsilon$$
              is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have
              $$U(P_{c},f)-L(P_{c},f)leq U(P,f)-L(P,f)<epsilon$$
              and you can proceed as you have done in your proof.



              An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.






              share|cite|improve this answer
























                2












                2








                2






                This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that
                $$U(P,f)-L(P,f)<epsilon$$
                is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have
                $$U(P_{c},f)-L(P_{c},f)leq U(P,f)-L(P,f)<epsilon$$
                and you can proceed as you have done in your proof.



                An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.






                share|cite|improve this answer












                This is essentially fine, although you are tacitly assuming that $c$ itself is a member of the partition, $P$. This isn't a big deal though - a $P$ such that
                $$U(P,f)-L(P,f)<epsilon$$
                is guaranteed by integrability, and you can always just add $c$ to this partition if it is not already there. Namely, if $P_{c}$ is this new partition, called a refinement of $P$, one must have
                $$U(P_{c},f)-L(P_{c},f)leq U(P,f)-L(P,f)<epsilon$$
                and you can proceed as you have done in your proof.



                An alternative approach is given by Lebesgue's Criterion for Riemann Integrability, which states that Riemann/Darboux Integrability is equivalent to boundedness plus continuity up to a null set (see Wikipedia for a good explanation of null sets). Since your functions is integrable on $[a,b]$, it is bounded and continuous up to a null set on $[a,b]$, and hence, on $[a,c]$ and $[c,b]$ as well.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 28 at 6:46









                JWP_HTX

                410313




                410313






























                    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.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • 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.


                    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%2f3016686%2friemann-darboux-integrability-of-subinterval%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